Buy BOOKS at Discounted Price

Method Of Integration

Class 12th Mathematics RS Aggarwal Solution
Exercise 13a
  1. integrate (2x+9)^{5} dx Evaluate the following integrals:
  2. Evaluate the following integrals:
  3. integrate root {3x-5}dx Evaluate the following integrals:
  4. integrate {1}/{ root {4x+3} } dx Evaluate the following integrals:…
  5. integrate {1}/{ root {3-4x} } dx Evaluate the following integrals:…
  6. integrate {1}/{ (2x-3)^{3/2} } dx Evaluate the following integrals:…
  7. integrate e^ { (2x-1) } dx Evaluate the following integrals:
  8. integrate e^ { (1-3x) } dx Evaluate the following integrals:
  9. integrate 3^ { (2-3x) } dx Evaluate the following integrals:
  10. integrate sin3xdx Evaluate the following integrals:
  11. integrate cos (5+6x) dx Evaluate the following integrals:
  12. integrate sinxroot {1+cos2x}dx Evaluate the following integrals:…
  13. integrate cosec^{2} (2x+5) dx Evaluate the following integrals:
  14. integrate sinxcosxdx Evaluate the following integrals:
  15. integrate sin^{3}xcosxdx Evaluate the following integrals:
  16. integrate ( root {cosx} ) sinxdx Evaluate the following integrals:…
  17. integrate {sin^{-1}x}/{ root { 1-x^{2} } } dx Evaluate the following…
  18. . integrate { sin (2tan^{-1}x) }/{ ( 1+x^{2} ) } dx . Evaluate the…
  19. integrate { cos (logx) }/{x}dx Evaluate the following integrals:…
  20. integrate { cosec^{2} (logx) }/{x}dx Evaluate the following integrals:…
  21. integrate {1}/{xlogx}dx Evaluate the following integrals:
  22. integrate { (x+1) (x+logx)^{2} }/{x}dx Evaluate the following integrals:…
  23. integrate { (logx)^{2} }/{x}dx Evaluate the following integrals:…
  24. integrate { cosroot {x} }/{ sqrt{x} } dx Evaluate the following integrals:…
  25. integrate e^{tanx} sec^{2}xdx Evaluate the following integrals:
  26. integrate e^{cos^{2}x} sin2xdx Evaluate the following integrals:…
  27. integrate sin (ax+b) cos (ax+b) dx Evaluate the following integrals:…
  28. integrate cos^{3}xdx Evaluate the following integrals:
  29. integrate {1}/{ x^{2} } e^{-1/x}dx Evaluate the following integrals:…
  30. integrate {1}/{ x^{2} } cos ( frac {1}/{x} ) dx Evaluate the following…
  31. integrate {dx}/{ ( e^{x} + e^{-x} ) } Evaluate the following integrals:…
  32. integrate { e^{2x} }/{ ( e^{2x} - 2 ) } dx Evaluate the following…
  33. integrate cotxlog (sinx) dx Evaluate the following integrals:
  34. integrate {cotx}/{ log (sinx) } dx Evaluate the following integrals:…
  35. integrate 2xsin ( x^{2} + 1 ) dx Evaluate the following integrals:…
  36. integrate secxlog (secx+tanx) dx Evaluate the following integrals:…
  37. integrate { tanroot {x} sec^{2}sqrt{x} }/{ sqrt{x} } dx Evaluate the…
  38. integrate { xtan^{-1}x^{2} }/{ ( 1+x^{4} ) } dx Evaluate the following…
  39. integrate { xsin^{-1}x^{2} }/{ root { 1-x^{4} } } dx Evaluate the…
  40. integrate {1}/{ ( root { 1-x^{2} } ) sin^{-1}x } dx Evaluate the following…
  41. integrate { root { (2+logx) } }/{x}dx Evaluate the following integrals:…
  42. integrate {sec^{2}x}/{ (1+tanx) } dx Evaluate the following integrals:…
  43. integrate {sinx}/{ (1+cosx) } dx Evaluate the following integrals:…
  44. integrate ( {1+tanx}/{1-tanx} ) dx Evaluate the following integrals:…
  45. i. integrate { (1+tanx) }/{ (x+logsecx) } dx ii. integrate { (1-sin2x)…
  46. integrate {sin2x}/{ ( a^{2} + b^{2}sin^{2}x ) } dx Evaluate the following…
  47. integrate {sin2x}/{ (a^{2}cos^{2}x+b^{2}sin^{2}x) } dx Evaluate the…
  48. integrate ( {2cosx-3sinx}/{3cosx+2sinx} ) dx Evaluate the following…
  49. integrate {4x}/{ ( 2x^{2} + 3 ) } dx Evaluate the following integrals:…
  50. integrate { (x+1) }/{ ( x^{2} + 2x-3 ) } dx Evaluate the following…
  51. integrate { (4x-5) }/{ ( 2x^{2} - 5x+1 ) } dx Evaluate the following…
  52. integrate { ( 9x^{2} - 4x+5 ) }/{ ( 3x^{3} - 2x^{2} + 5x+1 ) } dx Evaluate…
  53. integrate {secxcosecx}/{ log (tanx) } dx Evaluate the following integrals:…
  54. integrate { (1+cosx) }/{ (x+sinx)^{3} } dx Evaluate the following…
  55. integrate {sinx}/{ (1+cosx)^{2} } dx Evaluate the following integrals:…
  56. integrate { (2x+3) }/{ root { x^{2} + 3x-2 } } dx Evaluate the following…
  57. integrate { (2x-1) }/{ root { x^{2} - x-1 } } dx Evaluate the following…
  58. integrate {dx}/{ ( root {x+a} + sqrt{x+b} ) } Evaluate the following…
  59. integrate {dx}/{ ( root {1-3x} - sqrt{5-3x} ) } Evaluate the following…
  60. integrate { x^{2} }/{ ( 1+x^{6} ) } dx Evaluate the following integrals:…
  61. integrate { x^{3} }/{ ( 1+x^{8} ) } dx Evaluate the following integrals:…
  62. integrate {x}/{ ( 1+x^{4} ) } dx Evaluate the following integrals:…
  63. integrate { x^{5} }/{ root { 1+x^{3} } } dx Evaluate the following…
  64. integrate {x}/{ root {1+x} } dx Evaluate the following integrals:…
  65. integrate {1}/{ x root { x^{4} - 1 } } dx Evaluate the following…
  66. integrate x root {-1}dx Evaluate the following integrals:
  67. integrate (1-x) root {1+x}dx Evaluate the following integrals:
  68. integrate x root { x^{2} - 1 } dx Evaluate the following integrals:…
  69. integrate x root {3x-2}dx Evaluate the following integrals:
  70. integrate {dx}/{ xcos^{2} (1+logx) } Evaluate the following integrals:…
  71. integrate x^{2}sinx^{3} dx Evaluate the following integrals:
  72. integrate (2x+4) root { x^{2} + 4x+3 } dx Evaluate the following integrals:…
  73. integrate {sinx}/{ (sinx-cosx) } dx Evaluate the following integrals:…
  74. integrate {dx}/{ (1-tanx) } Evaluate the following integrals:
  75. integrate {dx}/{ (1-cotx) } Evaluate the following integrals:
  76. integrate {cos2x}/{ (sinx+cosx) } dx Evaluate the following integrals:…
  77. integrate { (cosx-sinx) }/{ (1+sin2x) } dx Evaluate the following…
  78. integrate { (x+1) (x+logx)^{2} }/{x}dx Evaluate the following integrals:…
  79. integrate xsin^{3}x^{2}cosx^{2} dx Evaluate the following integrals:…
  80. integrate {sec^{2}x}/{ root {1-tan^{2}x} } dx Evaluate the following…
  81. integrate e^{-x}cosec^{2} ( 2e^{-x} + 5 ) dx Evaluate the following…
  82. integrate 2xsec^{3} ( x^{2} + 3 ) tan ( x^{2} + 3 ) dx Evaluate the following…
  83. integrate {sin2x}/{ (a+bcosx)^{2} } dx Evaluate the following integrals:…
  84. integrate {dx}/{ (3-5x) } Evaluate the following integrals:
  85. integrate root {1+x}dx Evaluate the following integrals:
  86. integrate x^{2}e^ { x^{3} } cos ( e^ { x^{3} } ) dx Evaluate the following…
  87. integrate {e^{mtan^{-1}x}}/{ ( 1+x^{2} ) } dx Evaluate the following…
  88. integrate { (x+1) e^{x} }/{ cos^{2} ( xe^{x} ) } dx Evaluate the following…
  89. integrate { e^ { root {x} } cos ( e^ { sqrt{x} } ) }/{ sqrt{x} } dx…
  90. integrate root { e^{x} - 1 } dx Evaluate the following integrals:…
  91. Evaluate the following integrals:
  92. integrate { sec^{2} (2tan^{-1}x) }/{ ( 1+x^{2} ) } dx Evaluate the…
  93. ( {1+sin2x}/{x+sin^{2}x} ) dx Evaluate the following integrals:…
  94. integrate ( {1-tanx}/{x+logcosx} ) dx Evaluate the following integrals:…
  95. integrate { (1+cotx) }/{ (x+logsinx) } dx Evaluate the following integrals:…
  96. integrate {tanxsec^{2}x}/{ (1-tan^{2}x) } dx Evaluate the following…
  97. integrate { sin (2tan^{-1}x) }/{ ( 1+x^{2} ) } dx Evaluate the following…
  98. integrate {dx}/{ (x^{1/2}+x^{1/3}) } Evaluate the following integrals:…
  99. integrate (sin^{-1}x)^{2} dx Evaluate the following integrals:
  100. integrate { 2xtan^{-1}x^{2} }/{ ( 1+x^{4} ) } dx Evaluate the following…
  101. integrate { ( x^{2} + 1 ) }/{ ( x^{4} + 1 ) } dx Evaluate the following…
  102. integrate { (sinx+cosx) }/{ root {sin2x} } dx Evaluate the following…
Objective Questions I
  1. integrate (2x+3)^{5} dx = ? Mark (√) against the correct answer in each of the…
  2. integrate (3-5x)^{7} dx = ? Mark (√) against the correct answer in each of the…
  3. integrate {1}/{ (2-3x)^{4} } dx = ? Mark (√) against the correct answer in each of…
  4. integrate root {ax+b}dx = ? Mark (√) against the correct answer in each of the…
  5. integrate sec^{2} (7-4x) dx = ? Mark (√) against the correct answer in each of the…
  6. integrate cos3xdx = ? Mark (√) against the correct answer in each of the following:…
  7. integrate e^ { (5-3x) } dx = ? Mark (√) against the correct answer in each of the…
  8. integrate e^ { (3x+4) } dx = ? Mark (√) against the correct answer in each of the…
  9. integrate tan^{2} {x}/{2} dx = ? Mark (√) against the correct answer in each of the…
  10. integrate root {1-cosx}dx = ? Mark (√) against the correct answer in each of the…
  11. integrate root {1+sinx}dx = ? Mark (√) against the correct answer in each of the…
  12. integrate sin^{3}xdx = ? Mark (√) against the correct answer in each of the…
  13. integrate {logx}/{x}dx = ? Mark (√) against the correct answer in each of the…
  14. integrate { sec^{2} (logx) }/{x}dx = ? Mark (√) against the correct answer in each…
  15. integrate {1}/{ x (logx) } dx = ? Mark (√) against the correct answer in each of…
  16. integrate e^ { x^{3} } x^{2} dx = ? Mark (√) against the correct answer in each of…
  17. integrate { e^ { root {x} } }/{ sqrt{x} } dx = ? Mark (√) against the correct…
  18. integrate {e^{tan^{-1}x}}/{ ( 1+x^{2} ) } dx = ? Mark (√) against the correct…
  19. integrate { sinroot {x} }/{ sqrt{x} } dx = ? Mark (√) against the correct answer in…
  20. integrate ( root {sinx} ) cosxdx = ? Mark (√) against the correct answer in each of…
  21. integrate {1}/{ ( 1+x^{2} ) root {tan^{-1}x} } Mark (√) against the correct answer…
  22. integrate {cotx}/{ log (sinx) } dx = ? Mark (√) against the correct answer in each…
  23. integrate {1}/{ xcos^{2} (1+logx) } dx = ? Mark (√) against the correct answer in…
  24. integrate { x^{2}tan^{-1}x^{3} }/{ ( 1+x^{6} ) } dx = ? Mark (√) against the…
  25. integrate sec^{5}xtanxdx = ? Mark (√) against the correct answer in each of the…
  26. integrate cosec^{3} (2x+1) cot (2x+1) dx = ? Mark (√) against the correct answer in…
  27. integrate { tan (sin^{-1}x) }/{ root { 1-x^{2} } } dx = ? Mark (√) against the…
  28. integrate { tan (logx) }/{x}dx = ? Mark (√) against the correct answer in each of…
  29. integrate e^{x}cot ( e^{x} ) dx = ? Mark (√) against the correct answer in each of…
  30. integrate { e^{x} }/{ root { 1+e^{x} } } dx = ? Mark (√) against the correct…
  31. integrate {x}/{ root { 1-x^{2} } } dx = ? Mark (√) against the correct answer in…
  32. integrate { e^{x} (1+x) }/{ cos^{2} ( xe^{x} ) } dx = ? Mark (√) against the…
  33. integrate {dx}/{ ( e^{x} + e^{-x} ) } = ? Mark (√) against the correct answer in…
  34. integrate { 2^{x} }/{ 1-4^{x} } dx = ? Mark (√) against the correct answer in each…
  35. integrate {dx}/{ ( e^{x} - 1 ) } = ? Mark (√) against the correct answer in each of…
  36. integrate {1}/{ ( root {x}+x ) } dx = ? Mark (√) against the correct answer in each…
  37. integrate {dx}/{ (1+sinx) } = ? Mark (√) against the correct answer in each of the…
  38. integrate {sinx}/{ (1+sinx) } dx = ? Mark (√) against the correct answer in each of…
  39. integrate {sinx}/{ (1-sinx) } dx = ? Mark (√) against the correct answer in each of…
  40. integrate {dx}/{ (1+cosx) } = ? Mark (√) against the correct answer in each of the…
  41. integrate {dx}/{ (1-cosx) } = ? Mark (√) against the correct answer in each of the…
  42. integrate { 1-tan ( frac {x}/{2} ) }/{ 1+tan ( frac {x}/{2} ) } } dx = ? Mark (√)…
  43. integrate root { e^{x} } dx = ? Mark (√) against the correct answer in each of the…
  44. integrate {cosx}/{ (1+cosx) } dx = ? Mark (√) against the correct answer in each of…
  45. integrate sec^{2}xcosec^{2}xdx = ? Mark (√) against the correct answer in each of the…
  46. integrate { (1-cos2x) }/{ (1+cos2x) } dx = ? Mark (√) against the correct answer in…
  47. integrate { (1+cosx) }/{ (1-cosx) } dx = ? Mark (√) against the correct answer in…
  48. integrate {1}/{sin^{2}xcos^{2}x}dx = ? Mark (√) against the correct answer in each…
  49. integrate {cos2x}/{cos^{2}xsin^{2}x}dx = ? Mark (√) against the correct answer in…
  50. integrate { (cos2x-cos2alpha ) }/{ (cosx-cosalpha) } dx = ? Mark (√) against the…
  51. integrate tan^{-1} { root { {1-cos2x}/{1+cos2x} } } dx = ? Mark (√) against the…
  52. integrate tan^{-1} (secx+tanx) dx = ? Mark (√) against the correct answer in each of…
  53. integrate { (1+sinx) }/{ (1-sinx) } dx = ? Mark (√) against the correct answer in…
  54. integrate { x^{4} }/{ ( 1+x^{2} ) } dx = ? Mark (√) against the correct answer in…
  55. integrate { sin ( x - alpha ) }/{ sin ( x + alpha ) } dx = ? Mark (√) against the…
  56. integrate {1}/{ ( root {x+3} - sqrt{x+2} ) } dx = ? Mark (√) against the correct…
  57. integrate { (1+tanx) }/{ (1-tanx) } dx = ? Mark (√) against the correct answer in…
  58. integrate {dx}/{ x root { x^{6} - 1 } } = ? Mark (√) against the correct answer in…
  59. integrate { 3x^{2} }/{ ( 1+x^{8} ) } dx = ? Mark (√) against the correct answer in…
  60. integrate { (2x+1) root { x^{2} + x+1 } } dx = ? Mark (√) against the correct answer…
  61. integrate {dx}/{ { root {2x+3} + sqrt{2x+3} } } = ? Mark (√) against the correct…
  62. integrate tanxdx = ? Mark (√) against the correct answer in each of the following:…
  63. integrate secxdx = ? Mark (√) against the correct answer in each of the following:…
  64. integrate cosecxdx = ? Mark (√) against the correct answer in each of the following:…
  65. integrate { (1+sinx) }/{ (1+cosx) } dx = ? Mark (√) against the correct answer in…
  66. integrate {tanx}/{ (secx+cosx) } dx = ? Mark (√) against the correct answer in each…
  67. integrate root { {1+x}/{1-x} } dx = ? Mark (√) against the correct answer in each…
  68. integrate {1}/{ x^{2} } e^{-y/x}dx = ? Mark (√) against the correct answer in each…
  69. integrate { x^{3} }/{ ( 1+x^{8} ) } dx = ? Mark (√) against the correct answer in…
  70. integrate { (x+1) (x+logx)^{2} }/{x}dx = ? Mark (√) against the correct answer in…
  71. integrate { 2xtan^{-1}x^{2} }/{ ( 1+x^{4} ) } dx = ? Mark (√) against the correct…
  72. integrate {dx}/{ (2-3x) } = ? Mark (√) against the correct answer in each of the…
  73. integrate x root { x^{2} - 1 } dx = ? Mark (√) against the correct answer in each of…
  74. integrate e^ { (5-3x) } dx = ? Mark (√) against the correct answer in each of the…
  75. integrate e^{tanx} sec^{2}xdx = ? Mark (√) against the correct answer in each of the…
  76. integrate e^{cos^{2}x} sin2xdx = ? Mark (√) against the correct answer in each of the…
  77. integrate xsin^{3}x^{2}cosx^{2} dx = ? Mark (√) against the correct answer in each of…
  78. integrate { e^ { root {x} } cos ( e^ { sqrt{x} } ) }/{ sqrt{x} } dx = ? Mark (√)…
  79. integrate x^{2}sinx^{3} dx = ? Mark (√) against the correct answer in each of the…
  80. integrate { (x+1) e^{x} }/{ cos^{2} ( xe^{x} ) } dx = ? Mark (√) against the…
  81. integrate {1}/{ x root { x^{4} - 1 } } dx = ? Mark (√) against the correct answer…
  82. integrate x root {x-1}dx = ? Mark (√) against the correct answer in each of the…
  83. integrate x root { x^{2} - x } dx = ? Mark (√) against the correct answer in each of…
  84. integrate {dx}/{ ( 1 + root {x} ) } = ? Mark (√) against the correct answer in each…
  85. integrate root { e^{x} - 1 } dx Mark (√) against the correct answer in each of the…
  86. integrate {sinx}/{ (sinx-cosx) } dx = ? Mark (√) against the correct answer in each…
  87. integrate {dx}/{ (1-tanx) } = ? Mark (√) against the correct answer in each of the…
  88. integrate {dx}/{ (1-cotx) } = ? Mark (√) against the correct answer in each of the…
  89. integrate {sec^{2}x}/{ root {1-tan^{2}x} } dx = ? Mark (√) against the correct…
  90. integrate { ( x^{2} + 1 ) }/{ ( x^{4} + 1 ) } dx = ? Mark (√) against the correct…
  91. integrate {sin^{6}x}/{ cos^{8} } dx = ? Mark (√) against the correct answer in each…
  92. integrate sec^{5}xtanxdx = ? Mark (√) against the correct answer in each of the…
  93. integrate tan^{5}xdx = ? Mark (√) against the correct answer in each of the…
  94. integrate sin^{3}xcos^{3}xdx = ? Mark (√) against the correct answer in each of the…
  95. integrate sec^{4}xtanxdx = ? Mark (√) against the correct answer in each of the…
  96. integrate {logtanx}/{sinxcosx}dx = ? Mark (√) against the correct answer in each of…
  97. integrate sin^{3} (2x+1) dx = ? Mark (√) against the correct answer in each of the…
  98. integrate { root {tanx} }/{sinx+cosx}dx = ? Mark (√) against the correct answer in…
  99. integrate { (cos+sinx) }/{ (1-sin2x) } dx = ? Mark (√) against the correct answer…
  100. Q100 integrate root { e^{x} - 1 } dx = ? Mark (√) against the correct answer in each of…
  101. Q101 integrate {dx}/{ root {sin^{3}xcosx} } = ? Mark (√) against the correct answer in…
Exercise 13b
  1. (i) integrate sin^{2}xdx (ii) integrate cos^{2}xdx Evaluate the following…
  2. (i) integrate cos^{2} (x/2) dx (ii) integrate cot^{2} (x/2) dx Evaluate the…
  3. (i) integrate sin^{2}nxdx (ii) integrate sin^{5}xdx Evaluate the following…
  4. integrate cos^{3} (3x+5) dx Evaluate the following integrals:
  5. integrate sin^{7} (3-2x) dx Evaluate the following integrals:
  6. (i) ( {1-cos2x}/{1+cos2x} ) dx (ii) ( {1+cos2x}/{1-cos2x} ) dx Evaluate…
  7. (i) (ii) Evaluate the following integrals:
  8. integrate sin3xcos4xdx Evaluate the following integrals:
  9. integrate cos4xcos3xdx Evaluate the following integrals:
  10. integrate sin4xsin8xdx Evaluate the following integrals:
  11. integrate sin6xcosxdx Evaluate the following integrals:
  12. integrate sinxroot {1+cos2x}dx Evaluate the following integrals:…
  13. integrate cos^{4}xdx Evaluate the following integrals:
  14. integrate cos2xcos4xcos6xdx Evaluate the following integrals:
  15. integrate sin^{3}xcosxdx Evaluate the following integrals:
  16. integrate sec^{4}xdx Evaluate the following integrals:
  17. integrate cos^{3}xsin^{4}xdx Evaluate the following integrals:
  18. integrate cos^{4}xsin^{3}xdx Evaluate the following integrals:
  19. integrate sin^{2/3}xcos^{3}xdx Evaluate the following integrals:…
  20. integrate cos^{3/5}xsin^{3}xdx Evaluate the following integrals:…
  21. integrate cosec^{4}2xdx Evaluate the following integrals:
  22. integrate {cos2x}/{cosx}dx Evaluate the following integrals:
  23. integrate {cosx}/{ cos ( x + alpha ) } dx Evaluate the following…
  24. integrate cos^{3}xsin2xdx Evaluate the following integrals:
  25. integrate {cos^{9}x}/{sinx}dx Evaluate the following integrals:…
  26. integrate cos^{4}2xdx Evaluate the following integrals:
  27. integrate {sin^{2}x}/{ (1+cosx)^{2} } dx Evaluate the following integrals:…
  28. integrate {dx}/{ (3cosx+4sinx) } Evaluate the following integrals:…
  29. integrate {dx}/{ (acosx+bsinx)^{2} } a 0 and b 0 Evaluate the…
  30. integrate {dx}/{ (cosx-sinx) } Evaluate the following integrals:…
  31. integrate (2tanx-3cotx)^{2} dx Evaluate the following integrals:…
  32. integrate sinxsin2xsin3xdx Evaluate the following integrals:
  33. integrate ( {1-cotx}/{1+cotx} ) dx Evaluate the following integrals:…
  34. integrate {dx}/{ (2sinx+cosx+3) } Evaluate the following integrals:…
Exercise 13c
  1. integrate xe^{x} dx Evaluate the following integrals:
  2. integrate xcosxdx Evaluate the following integrals:
  3. integrate xe^{2x} dx Evaluate the following integrals:
  4. integrate xsin3xdx Evaluate the following integrals:
  5. integrate xcos2xdx Evaluate the following integrals:
  6. integrate xlog2xdx Evaluate the following integrals:
  7. integrate xcosec^{2}xdx Evaluate the following integrals:
  8. integrate x^{2}cosxdx Evaluate the following integrals:
  9. integrate xsin^{2}xdx Evaluate the following integrals:
  10. integrate xtan^{2}xdx Evaluate the following integrals:
  11. integrate x^{2}e^{x} dx Evaluate the following integrals:
  12. integrate x^{2}cos^{3}xdx Evaluate the following integrals:
  13. integrate x^{2}e^{3x} dx Evaluate the following integrals:
  14. integrate x^{2}sin^{2}xdx Evaluate the following integrals:
  15. integrate x^{3}log2xdx Evaluate the following integrals:
  16. integrate x c. log (x+1) dx Evaluate the following integrals:
  17. integrate {logx}/{ x^{n} } dx Evaluate the following integrals:…
  18. integrate 2x^{3}e^ { x^{2} } dx Evaluate the following integrals:…
  19. integrate xsin^{3}xdx Evaluate the following integrals:
  20. integrate xcos^{3}xdx Evaluate the following integrals:
  21. integrate x^{3}cosx^{2} dx Evaluate the following integrals:
  22. Evaluate the following integrals:
  23. integrate xsinxcosxdx Evaluate the following integrals:
  24. integrate cosroot {x}dx Evaluate the following integrals:
  25. integrate cosec^{3}xdx Evaluate the following integrals:
  26. integrate xsin^{3}xcosxdx Evaluate the following integrals:
  27. integrate sinxlog (cosx) dx Evaluate the following integrals:
  28. integrate { log (logx) }/{x}dx Evaluate the following integrals:…
  29. integrate log ( 2+x^{2} ) dx Evaluate the following integrals:
  30. integrate {x}/{ (1+sinx) } dx Evaluate the following integrals:…
  31. integrate { {1}/{logx} - frac {1}/{ (logx)^{2} } } dx Evaluate the following integrals:…
  32. integrate e^{-x}cos2xcos4xdx Evaluate the following integrals:
  33. integrate e^ { root {x} } dx Evaluate the following integrals:
  34. integrate e^{sinx} sin2xdx Evaluate the following integrals:
  35. integrate {xsin^{-1}x}/{ root { 1-x^{2} } } dx Evaluate the following integrals:…
  36. integrate {x^{2}tan^{-1}x}/{ ( 1+x^{2} ) } dx Evaluate the following integrals:…
  37. integrate { log (x+2) }/{ (x+2)^{2} } dx Evaluate the following integrals:…
  38. integrate xsin^{-1}xdx Evaluate the following integrals:
  39. integrate xcos^{-1}xdx Evaluate the following integrals:
  40. integrate cot^{-1}xdx Evaluate the following integrals:
  41. integrate xcot^{-1}xdx Evaluate the following integrals:
  42. integrate x^{2}cot^{-1}xdx [CBSE 2006C] Evaluate the following integrals:…
  43. integrate sin^{-1}root {x}dx Evaluate the following integrals:
  44. integrate cos^{-1}root {x}dx Evaluate the following integrals:
  45. integrate cos^{-1} ( 4x^{3} - 3x ) dx Evaluate the following integrals:…
  46. integrate cos^{-1} ( { 1-x^{2} }/{ 1+x^{2} } ) dx Evaluate the following…
  47. integrate tan^{-1} ( {2x}/{ 1-x^{2} } ) dx Evaluate the following…
  48. integrate tan^{-1} ( { 3x-x^{3} }/{ 1-3x^{2} } ) dx Evaluate the following…
  49. integrate {sin^{-1}x}/{ x^{2} } dx Evaluate the following integrals:…
  50. integrate {tanxsec^{2}x}/{ (1-tan^{2}x) } dx Evaluate the following…
  51. integrate e^{3x}sin4xdx Evaluate the following integrals:
  52. integrate e^{2x}sinxdx Evaluate the following integrals:
  53. integrate e^{2x}sinxcosxdx Evaluate the following integrals:
  54. integrate e^{2x}cos (3x+4) dx Evaluate the following integrals:
  55. integrate e^{-x}cosxdx Evaluate the following integrals:
  56. integrate e^{x} (sinx+cosx) dx Evaluate the following integrals:…
  57. integrate e^{x} (cotx-cosec^{2}x) dx Evaluate the following integrals:…
  58. integrate e^{x}secx (1+tanx) dx Evaluate the following integrals:…
  59. integrate e^{x} ( tan^{-1}x + {1}/{ 1+x^{2} } ) dx Evaluate the following…
  60. integrate e^{x} (cotx+logsinx) dx Evaluate the following integrals:…
  61. integrate e^{x} (tanx-logcosx) dx Evaluate the following integrals:…
  62. integrate e^{x} [ secx+log (secx+tanx) ]dx Evaluate the following integrals:…
  63. integrate e^{x} ( {1+sinxcosx}/{cos^{2}x} ) dx Evaluate the following…
  64. integrate e^{x} ( {sinxcosx-1}/{sin^{2}x} ) dx Evaluate the following…
  65. integrate e^{x} ( {cosx+sinx}/{cos^{2}x} ) dx Evaluate the following…
  66. integrate e^{x} ( {2-sin2x}/{1-cos2x} ) dx Evaluate the following…
  67. integrate e^{x} ( {1+sinx}/{1+cosx} ) dx Evaluate the following integrals:…
  68. integrate e^{x} ( {sin4x-4}/{1-cos4x} ) dx Evaluate the following…
  69. integrate { e^{x} [ root { 1-x^{2} } sin^{-1}x+1] }/{ sqrt { 1-x^{2} } }…
  70. integrate e^{x} ( {1+xlogx}/{x} ) dx Evaluate the following integrals:…
  71. integrate e^{x} c. {x}/{ (1+x)^{2} } dx Evaluate the following integrals:…
  72. integrate e^{x} { (x-1) }/{ (x+1)^{3} } dx Evaluate the following…
  73. integrate e^{x} { (2-x) }/{ (1-x)^{2} } dx Evaluate the following…
  74. integrate e^{x} c. { (x-3) }/{ (x-1)^{3} } dx Evaluate the following…
  75. integrate e^{3x} ( {3x-1}/{ 9x^{2} } ) dx Evaluate the following integrals:…
  76. integrate { (x+1) }/{ (x+2)^{2} } e^{x} dx Evaluate the following…
  77. integrate { xe^{2x} }/{ (1+2x)^{2} } dx Evaluate the following integrals:…
  78. integrate e^{2x} ( {2x-1}/{ 4x^{2} } ) dx Evaluate the following integrals:…
  79. integrate e^{x} ( logx + {1}/{ x^{2} } ) dx Evaluate the following…
  80. integrate {logx}/{ (1+logx)^{2} } dx Evaluate the following integrals:…
  81. integrate { sin (logx) + cos (logx) } dx Evaluate the following integrals:…
  82. integrate { {1}/{logx} - frac {1}/{ (logx)^{2} } } dx Evaluate the…
  83. integrate { log (logx) + {1}/{ (logx)^{2} } } dx Evaluate the following…
  84. integrate ( { sin^{-1}root {x}-cos^{-1}sqrt{x} }/{
  85. integrate 5^{5x} c. 5^ { 5^{x} } 5^{x} dx Evaluate the following…
  86. integrate e^{2x} ( {1+sin2x}/{1+cos2x} ) dx Evaluate the following…
  87. integrate e^{2x} ( {1-sin2x}/{1-cos2x} ) dx Evaluate the following…
Objective Questions Ii
  1. integrate xe^{x} dx = ? Mark (√) against the correct answer in each of the following:…
  2. integrate xe^{2x} dx = ? Mark (√) against the correct answer in each of the following:…
  3. integrate xcos2xdx = ? Mark (√) against the correct answer in each of the following:…
  4. integrate xsec^{2}xdx = ? Mark (√) against the correct answer in each of the…
  5. integrate xsin2xdx = ? Mark (√) against the correct answer in each of the following:…
  6. integrate xlogxdx = ? Mark (√) against the correct answer in each of the following:…
  7. integrate xcosec^{2}xdx = ? Mark (√) against the correct answer in each of the…
  8. integrate xsinxcosxdx = ? Mark (√) against the correct answer in each of the…
  9. integrate xcos^{2}xdx = ? Mark (√) against the correct answer in each of the…
  10. integrate {logx}/{ x^{2} } dx = ? Mark (√) against the correct answer in each of…
  11. integrate logxdx = ? Mark (√) against the correct answer in each of the following:…
  12. integrate log_{10}xdx = ? Mark (√) against the correct answer in each of the…
  13. integrate (logx)^{2} dx = ? Mark (√) against the correct answer in each of the…
  14. integrate e^ { root {x} } dx = ? Mark (√) against the correct answer in each of the…
  15. integrate cosroot {x}dx = ? Mark (√) against the correct answer in each of the…
  16. integrate cos (logx) dx = ? Mark (√) against the correct answer in each of the…
  17. integrate sec^{3}xdx = ? Mark (√) against the correct answer in each of the…
  18. integrate { {1}/{ (logx) } - frac {1}/{ (logx)^{2} } } dx = ? Mark (√) against the…
  19. integrate 2x^{3}e^ { x^{2} } dx = ? Mark (√) against the correct answer in each of…
  20. integrate ( x2^{x} ) dx = ? Mark (√) against the correct answer in each of the…
  21. integrate xcot^{2}xdx = ? Mark (√) against the correct answer in each of the…
  22. integrate sinroot {x}dx = ? Mark (√) against the correct answer in each of the…
  23. integrate e^{sinx} sin2xdx = ? Mark (√) against the correct answer in each of the…
  24. integrate {sin^{-1}x}/{ ( 1-x^{2} ) ^{3/2} } dx = ? Mark (√) against the correct…
  25. integrate {xtan^{-1}x}/{ ( 1-x^{2} ) ^{3/2} } dx = ? Mark (√) against the correct…
  26. integrate xtan^{-1}xdx = ? Mark (√) against the correct answer in each of the…
  27. integrate tan^{-1}root {x}dx = ? Mark (√) against the correct answer in each of the…
  28. integrate cos^{-1}xdx = ? Mark (√) against the correct answer in each of the…
  29. integrate tan^{-1}xdx = ? Mark (√) against the correct answer in each of the…
  30. integrate sec^{-1}xdx = ? Mark (√) against the correct answer in each of the…
  31. integrate sin^{-1} ( 3x-4x^{3} ) dx = ? Mark (√) against the correct answer in each…
  32. integrate sin^{-1} ( {2x}/{ 1+x^{2} } ) dx = ? Mark (√) against the correct answer…
  33. integrate tan^{-1}root { {1-x}/{1+x} } dx = ? Mark (√) against the correct answer…
  34. integrate tan^{-1} ( { 3x-x^{3} }/{ 1-3x^{2} } ) dx = ? Mark (√) against the…
  35. integrate x^{2}cosxdx = ? Mark (√) against the correct answer in each of the…
  36. integrate sinxlog (cosx) dx = ? Mark (√) against the correct answer in each of the…
  37. integrate xsinxcosxdx = ? Mark (√) against the correct answer in each of the…
  38. integrate x^{3}cosx^{2} dx = ? Mark (√) against the correct answer in each of the…
  39. integrate cos^{-1} ( { 1-x^{2} }/{ 1+x^{2} } ) dx = ? Mark (√) against the correct…
  40. integrate xtan^{-1}xdx = ? Mark (√) against the correct answer in each of the…
  41. integrate sin (logx) dx = ? Mark (√) against the correct answer in each of the…
  42. integrate (sin^{-1}x)^{2} dx = ? Mark (√) against the correct answer in each of the…
  43. integrate e^{x} { {1}/{x} - frac {1}/{ x^{2} } } dx = ? Mark (√) against the…
  44. integrate e^{x} ( {1}/{ x^{2} } - frac {2}/{ x^{3} } ) dx = ? Mark (√) against the…
  45. integrate e^{x} { sin^{-1}x + {1}/{ root { 1-x^{2} } } } dx = ? Mark (√) against…
  46. integrate e^{x} (tanx+logsecx) dx = ? Mark (√) against the correct answer in each of…
  47. integrate e^{x} (tanx+logsecx) dx = ? Mark (√) against the correct answer in each of…
  48. integrate e^{x} (cotx+logsinx) dx = ? Mark (√) against the correct answer in each of…
  49. integrate e^{x} { tan^{-1}x + {1}/{ ( 1+x^{2} ) } } dx = ? Mark (√) against the…
  50. integrate e^{x} (tanx-logcosx) dx = ? Mark (√) against the correct answer in each of…
  51. integrate e^{x} (cotx-cosec^{2}x) dx = ? Mark (√) against the correct answer in each…
  52. integrate e^{x} (sinx+cosx) dx = ? Mark (√) against the correct answer in each of the…
  53. integrate e^{x}secx (1+tanx) dx = ? Mark (√) against the correct answer in each of…
  54. integrate e^{x} ( {1+xlogx}/{x} ) dx = ? Mark (√) against the correct answer in…
  55. integrate e^{x} c. {x}/{ (1+x)^{2} } dx = ? Mark (√) against the correct answer in…
  56. integrate e^{x} ( {1+sinx}/{1+cosx} ) dx = ? Mark (√) against the correct answer in…
Exercise 13a
Question 1.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2x + 9 = t ⇒ 2 dx = dt
Question 2.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 7 – 3x = t ⇒ -3 dx = dt
Question 3.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 3x - 5 = t ⇒ 3 dx = dt
Question 4.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 4x + 3 = t ⇒ 4 dx = dt
Question 5.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 3 – 4x = t ⇒ -4 dx = dt
Question 6.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2x – 3 = t ⇒ 2 dx = dt
Question 7.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2x – 1 = t ⇒ 2 dx = dt
Question 8.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 1 – 3x = t ⇒ -3 dx = dt
Question 9.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2 – 3x = t ⇒ -3 dx = dt
Question 10.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 3x = t ⇒ 3 dx = dt
Question 11.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 5 + 6x = t ⇒ 6 dx = dt
Question 12.

Evaluate the following integrals:
Answer:

Formula



Therefore ,




Put sin x =t ⇒ cos x dx = dt
Question 13.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put 2x + 5 =t ⇒ 2 dx = dt
Question 14.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put sin x =t ⇒ cos x dx = dt
Question 15.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put sin x =t ⇒ cos x dx = dt
Question 16.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put cos x =t ⇒ -sin x dx = dt
Question 17.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put =t ⇒ dx = dt
Question 18.

Evaluate the following integrals:

..
Answer:

Formula


Therefore ,


Put ⇒





Question 19.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put ⇒





Question 20.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put ⇒





Question 21.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put ⇒





Question 22.

Evaluate the following integrals:
Answer:

Formula



Therefore ,


Put ⇒





Question 23.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put ⇒





Question 24.

Evaluate the following integrals:
Answer:

Formula


Therefore ,


Put ⇒





Question 25.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put tan x = t ⇒ dx = dt
Question 26.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒ dx = dt
Question 27.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put ax+b = t ⇒ adx = dt



Put sin t = z � cos t dt = dz
Question 28.

Evaluate the following integrals:
Answer:

Formula =



Therefore ,
Question 29.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒





Question 30.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒





Question 31.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,



Put = t ⇒





Question 32.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒





Question 33.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put log (sin x) = t ⇒ �





Question 34.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put log (sin x) = t ⇒ �





Question 35.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put x2 + 1= t ⇒





Question 36.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put log (sec x + tan x)= t




Sec x dx = dt
Question 37.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


= t
Question 38.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒ �





Question 39.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒ �





Question 40.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put = t ⇒ �





Question 41.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2 + log x = t ⇒





Question 42.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 1 + tan x = t ⇒





Question 43.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 1 + cos x = t ⇒





Question 44.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,



Put cos x - sin x = t ⇒ (- cos x - sin x) dx = dt
Question 45.

Evaluate the following integrals:

i.

ii.


Answer:

(i)


Formula =


Therefore ,


Put x + log (sec x) = t ⇒





(ii)


Formula =


Therefore ,


Put = t ⇒
Question 46.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put �






Question 47.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put







Question 48.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 3cos x + 2sin x = t ⇒ (2cos x - 3sin x) dx = dt
Question 49.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put 2x2 +3= t ⇒ (4x) dx = dt
Question 50.

Evaluate the following integrals:
Answer:

Formula =


Therefore ,


Put x2+2x+3= t ⇒ (2x+2) dx = dt � 2(x+1)dx=dt
Question 51.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let 2x2 - 5x + 1 = t




⇒ (4x - 5)dx = dt


Putting this value in equation (i)





Ans)



Question 52.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let 3x3 - 2x2 + 5x + 1 = t




⇒ (9x2 - 4x + 5)dx = dt


Putting this value in equation (i)





Ans)



Question 53.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t






⇒ ()dx = dt


Putting this value in equation (i)





Ans)



Question 54.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t





⇒ ()dx = dt


Putting this value in equation (i)
Question 55.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t





⇒ ()dx = dt


Putting this value in equation (i)
Question 56.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t



⇒ () dx = dt


Putting this value in equation (i)
Question 57.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t





⇒ () dx = dt


Putting this value in equation (i)
Question 58.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)
Question 59.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)
Question 60.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)



Let t





Putting this value in equation (i)
Question 61.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)



Let t





Putting this value in equation (i)
Question 62.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)



Let t





Putting this value in equation (i)
Question 63.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t


⇒ t - 1




⇒ dx =


Putting this value in equation (i)
Question 64.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t


⇒ t - 1


⇒ dx =


Putting this value in equation (i)
Question 65.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Multiplying numerator and denominator with x



Let t



⇒ xdx =


Putting this value in equation (i)
Question 66.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t


x = t + 1



Putting this value in equation (i)
Question 67.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t


x = t - 1



Putting this value in equation (i)
Question 68.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t




Putting this value in equation (i)
Question 69.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t






Putting this value in equation (i)
Question 70.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t




Putting this value in equation (i)
Question 71.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t




Putting this value in equation (i)
Question 72.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let t




Putting this value in equation (i)
Question 73.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)






Let t




Putting this value in equation (i)
Question 74.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)








Let = t




Putting this value in equation (i)
Question 75.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)








Let = t




Putting this value in equation (i)
Question 76.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)






Let = t




Putting this value in equation (i)
Question 77.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)





Let = t




Putting this value in equation (i)
Question 78.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 79.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 80.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 81.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 82.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t





Putting this value in equation (i)
Question 83.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used: (i)


(ii)


We have, … (i)



Let = t




Putting this value in equation (i)
Question 84.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 85.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t



Putting this value in equation (i)
Question 86.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 87.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 88.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 89.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 90.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t2


= t2


t2 + 1





Putting this value in equation (i)
Question 91.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)



Let = t




Putting this value in equation (i)
Question 92.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 93.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 94.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t





Putting this value in equation (i)
Question 95.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t





Putting this value in equation (i)
Question 96.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t





Putting this value in equation (i)
Question 97.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t





Putting this value in equation (i)
Question 98.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used: (i)


(ii)


We have, … (i)


Let




Putting this value in equation (i)
Question 99.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t , x = sint ,








Putting this value in equation (i)
Question 100.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let = t




Putting this value in equation (i)
Question 101.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Dividing Numerator and Denominator by x2,





Let = t



Putting this value in equation (i)
Question 102.

Evaluate the following integrals:
Answer:

To find: Value of


Formula used:


We have, … (i)


Let (sinx – cosx) = t




⇒ t2 = sin2x - 2sinx. cosx + cos2x


⇒ t2 = 1 - 2sinx.cosx


⇒ 2sinx.cosx = 1 - t2


⇒ sin2x = 1 - t2


Putting this value in equation (i)





Let


⇒ … (ii)



Now if


Then cosθ =


⇒ cosθ =


⇒ cosθ =


⇒ cosθ =


Now tanθ =


Now tanθ =



Comparing the value θ from eqn. (ii)



Dividing Numerator and denominator from cosx



Ans.)



Objective Questions I
Question 1.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, 2x + 3 = z


⇒ 2dx = dz


So,


where c is the integrating constant.
Question 2.

Mark (√) against the correct answer in each of the following:



A. -5(3 – 5x)6 + C

B.

C.

D. none of these
Answer:

Given =


Let, 3 – 5x = z


⇒ -5dx = dz


So,


where c is the integrating constant.
Question 3.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, 2 – 3x = z


⇒ -3dx = dz


So,


where c is the integrating constant.
Question 4.

Mark (√) against the correct answer in each of the following:



A.
B.

C.

D. none of these
Answer:

Given =


Let, ax + b = z2


⇒ adx = 2zdz


So,


where c is the integrating constant.
Question 5.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 4 tan (7 – 4x) + C

D. - 4 tan (7 – 4x) + C
Answer:

Given =


Let, 7 – 4x = z


⇒ -4dx = dz


So,


where c is the integrating constant.
Question 6.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 3 sin 3x + C

D. -3 sin 3x + C
Answer:

Given =


So, where c is the integrating constant.
Question 7.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, 5 – 3x = z


⇒ -3dx = dz


So,


where c is the integrating constant.
Question 8.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, 3x + 4 = z


⇒ 3dx = dz


So,



where c is the integrating constant.
Question 9.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


Let,


⇒ dx = 2dz


So,



where c is the integrating constant.
Question 10.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


So,



Let 1 + cosx = u2


So, -sinxdx = 2udu



where c is the integrating constant.
Question 11.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


So,



Let 1 - sinx = u2


So, -cosxdx = 2udu



where c is the integrating constant.
Question 12.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


So,



Let cosx = u


So, -sinxdx = du



where c is the integrating constant.
Question 13.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


Let, logx = u


So,


So,



where c is the integrating constant.
Question 14.

Mark (√) against the correct answer in each of the following:



A. log (tan x) + C

B. - log (tan x) + C

C. tan (tan x) + C

D. - tan (log x) + C
Answer:

Given =


Let, logx = z


⇒


So,



where c is the integrating constant.
Question 15.

Mark (√) against the correct answer in each of the following:



A.

B.

C. (log x)2 + C

D. log |log x| + C
Answer:

Given =


Let, logx = z


⇒


So,



where c is the integrating constant.
Question 16.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, x3 = z


⇒ 3x2dx = dz


⇒


So,



where c is the integrating constant.
Question 17.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, x = z2


⇒ dx = 2zdz


So,



where c is the integrating constant.
Question 18.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, tan-1x = z


⇒


So,



where c is the integrating constant.
Question 19.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


Let, x = z2


⇒ dx = 2zdz


So,



where c is the integrating constant.
Question 20.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


Let, sinx = z2


⇒ cosxdx = 2zdz


So,



where c is the integrating constant.
Question 21.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, tan-1x = z2


⇒


So,



where c is the integrating constant.
Question 22.

Mark (√) against the correct answer in each of the following:



A. log |cot x| + C

B. log |cot x cosec x| + C

C. log |log sin x| + C

D. none of these
Answer:

Given =


Let, sinx = z


⇒ cosxdx = dz


So,



Let, logz = u


⇒


So,



where c is the integrating constant.
Question 23.

Mark (√) against the correct answer in each of the following:



A. tan (1 + log x) + C

B. cot (1 + log x) + C

C. sec (1 + log x) + C

D. none of these
Answer:

Given =


Let, 1 + logx = z


⇒


So,



where c is the integrating constant.
Question 24.

Mark (√) against the correct answer in each of the following:



A.

B. log |tan-1 x3| + C

C.

D. none of these
Answer:

Given =


Let, tan-1x3 = z


⇒


⇒


So,



where c is the integrating constant.
Question 25.

Mark (√) against the correct answer in each of the following:



A. 5 tan5 x + C

B.

C. 5 log |cos x| + C

D. none of these
Answer:

Given =


So,


Let, secx = z


⇒ secxtanxdx = dz



where c is the integrating constant.
Question 26.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


So,



Let, cosec(2x + 1) = z


⇒ -2cosec(2x + 1)cot(2x + 1)dx = dz



where c is the integrating constant.
Question 27.

Mark (√) against the correct answer in each of the following:



A. log |sec (sin-1 x)| + C

B. log |cos (sin-1 x)| + C

C. tan (sin1 x) + C

D. none of these
Answer:

Given =


Let, sin-1x = z


⇒


So,



where c is the integrating constant.
Question 28.

Mark (√) against the correct answer in each of the following:



A. x tan (log x) + C

B. log |tan x| + C

C. log |cos (log x)| + C

D. - log |cos (log x)| + C
Answer:

Given =


Let, logx = z


⇒


So,



where c is the integrating constant.
Question 29.

Mark (√) against the correct answer in each of the following:



A. cot (ex) + C

B. log |sin ex| + C

C. log |cosec ex| + C

D. none of these
Answer:

Given =


Let, ex = z


⇒ exdx = dz


So,



where c is the integrating constant.
Question 30.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =


Let, 1 + ex = z2


⇒ exdx = 2zdz


So,



where c is the integrating constant.
Question 31.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given =


Let, 1 – x2 = z2


⇒ -2xdx = 2zdz


So,



where c is the integrating constant.
Question 32.

Mark (√) against the correct answer in each of the following:



A. tan (xex) + C

B. cot (xex) + C

C. exx tan x + C

D. none of these
Answer:

Given =


Let, xex = z


⇒ ex(1 + x)dx = dz


So,



where c is the integrating constant.
Question 33.

Mark (√) against the correct answer in each of the following:



A. cot-1 (ex) + C

B. tan-1 (ex) + C

C. log |ex + 1| + C

D. none of these
Answer:

Given =



Let, ex + 1 = z


⇒ exdx = dz


So,



where c is the integrating constant.
Question 34.

Mark (√) against the correct answer in each of the following:



A. sin-1 (2x) + C

B. (log e2) sin-1 (2x) + C

C. (log e2) cos-1 (2x) + C

D. log2 e) sin-1 (2x) + C
Answer:

Given =



Let, 2x = z


⇒ 2x(log2)dx = dz


So,



where c is the integrating constant.
Question 35.

Mark (√) against the correct answer in each of the following:



A. log |ex – 1| + C

B. log |1 – e-x| + C

C. log |ex – 1| + C

D. none of these
Answer:

Given =



Let, ex – 1 = z


⇒ exdx = dz


So,



where c is the integrating constant.
Question 36.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given =



Let,


⇒


So,



where c is the integrating constant.
Question 37.

Mark (√) against the correct answer in each of the following:



A. tan x + sec x + C

B. tan x – sec x + C

C.

D. none of these
Answer:

Given



Let,


⇒


So,



where c is the integrating constant.
Question 38.

Mark (√) against the correct answer in each of the following:



A. x + tan x – sec x + C

B. x – tan x – sec x + C

C. x – tan x + sec x + C

D. none of these
Answer:

Given




Let,


⇒


So,



where c is the integrating constant.
Question 39.

Mark (√) against the correct answer in each of the following:



A. - x + sec x – tan x + C

B. x + cos x – sin x + C

C. - log |1 – sin x| + C

D. none of these
Answer:

Given




Let,


⇒


So,



where c is the integrating constant.
Question 40.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given



where c is the integrating constant.
Question 41.

Mark (√) against the correct answer in each of the following:



A.

B. log |x – sin x| + C

C.

D.


Answer:

Given



where c is the integrating constant.
Question 42.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Given



Let,


⇒


So,



where c is the integrating constant.
Question 43.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given



where c is the integrating constant.
Question 44.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given



[From Question no. 40] where c is the integrating constant.
Question 45.

Mark (√) against the correct answer in each of the following:



A. tan x – cot x + C

B. tan x + cot x + C

C. - tan x + cot x + C

D. none of these
Answer:

Given




where c is the integrating constant.
Question 46.

Mark (√) against the correct answer in each of the following:



A. tan x + x + C

B. tan x – x + C

C. - tan x + x + C

D. none of these
Answer:

Given



where c is the integrating constant.
Question 47.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Given



where c is the integrating constant.
Question 48.

Mark (√) against the correct answer in each of the following:



A. tan x + cot x + C

B. tan x – cot x + C

C. - tan x + cot x + C

D. none of these
Answer:

Given




where c is the integrating constant.
Question 49.

Mark (√) against the correct answer in each of the following:



A. cot x + tan x + C

B. - cot x + tan x + C

C. cot x – tan x + C

D. - cot x – tan x + C
Answer:

Given




where c is the integrating constant.
Question 50.

Mark (√) against the correct answer in each of the following:



A. sin x + x cos α + C

B. 2sin x + x cos α + C

C. 2 sin x + 2x cos α + C

D. none of these
Answer:

Given





where c is the integrating constant.
Question 51.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒


⇒
Question 52.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-



Therefore ,


⇒


⇒


⇒


(Multiply by in numerator and denominator)


⇒


⇒


Question 53.

Mark (√) against the correct answer in each of the following:



A. 2 tan x + x – 2sec x + C

B. 2 tan x – x + 2 sec x + C

C. 2 tan x – x – 2sec x + C

D. none of these
Answer:

Formula :-


Therefore ,


⇒


⇒


⇒


⇒


⇒


⇒


Put cos x = t


Therefore -> sin x dx = - dt


⇒


⇒


⇒
Question 54.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒


⇒


⇒


⇒
Question 55.

Mark (√) against the correct answer in each of the following:



A. x cos 2α - sin 2α . log |sin (x + α)| + C

B. x cos 2α + sin 2α . log|sin (x + α)| + C

C. x cos 2α + sin α . log |sin (x + α)| + C

D. none of these
Answer:

Formula :-




Therefore ,


⇒


⇒


⇒


⇒
Question 56.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-




Therefore ,


⇒ (Rationalizing the denominator)


⇒


⇒


⇒
Question 57.

Mark (√) against the correct answer in each of the following:



A. - log |cos x – sin x| + C

B. log |cos x – sin x| + C

C. log |cos x + sin x | + C

D. none of these
Answer:

Formula :-




Therefore ,


⇒ (Rationalizing the denominator)


⇒


Put cos x - sin x = t


(- sin x - cos x) dx = dt


(sin x + cos x) dx = -dt


⇒


⇒
Question 58.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put


⇒


⇒


⇒


⇒
Question 59.

Mark (√) against the correct answer in each of the following:



A. sin-1 x3 + C

B. cos-1 x3 + C

C. tan-1 x3 + C

D. cot-1 x3 + C
Answer:

Formula :-


Therefore ,


Put


⇒


⇒


⇒
Question 60.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put


⇒


⇒


⇒


Question 61.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-




Therefore ,


⇒ (Rationalizing the denominator)


⇒


⇒


⇒


⇒
Question 62.

Mark (√) against the correct answer in each of the following:



A. log |cos x| + C

B. - log |cos x| + C

C. log |sin x| + C

D. - log |sin x| + C
Answer:

Formula :-




Therefore ,


⇒


Put cos x = t -sin x dx = dt


⇒


⇒


⇒
Question 63.

Mark (√) against the correct answer in each of the following:



A. log |sec x - tan x| + C

B. - log |sec x + tan x| + C

C. log |sec x + tan x| + C

D. none of these
Answer:

Formula :-




Therefore ,


⇒


⇒


Put , (dx = dt


⇒


⇒
Question 64.

Mark (√) against the correct answer in each of the following:



A. log |cosec x - cot x| + C

B. - log |cosec x - cot x| + C

C. log |cosec x + cot x| + C

D. none of these
Answer:

Formula :-




Therefore ,




Put , dx = dt
Question 65.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,
Question 66.

Mark (√) against the correct answer in each of the following:



A. tan-1 (cos x) + C

B. - tan-1 (cos x) + C

C. cot-1 (cos x) + C

D. none of these
Answer:

Formula :-


Therefore ,



Put sec x = t (sec x tan x) dx = dt
Question 67.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,




Put -2x dx = dt
Question 68.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put





Question 69.

Mark (√) against the correct answer in each of the following:



A. tan-1 x4 + C

B. 4 tan-1 x4 + C

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put





Question 70.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒





Question 71.

Mark (√) against the correct answer in each of the following:



A. (tan-1x2)2 + C

B. 2 tan-1 x2 + C

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒





Question 72.

Mark (√) against the correct answer in each of the following:



A. - 3 log |2 – 3x| + C

B.

C. - log |2 – 3x| + C

D. none of these
Answer:

Formula :-


Therefore ,


Put





Question 73.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put



⇒



Question 74.

Mark (√) against the correct answer in each of the following:



A.

B.

C. -3(5 – 3x) log 3 + C

D. none of these
Answer:

Formula :-


Therefore ,


Put



⇒



Question 75.

Mark (√) against the correct answer in each of the following:



A. etan x + tan x + C

B. etan x . tan x + C

C. etan x + C

D. none of these
Answer:

Formula :-


Therefore ,


Put



⇒


Question 76.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒ ⇒



⇒


Question 77.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒



⇒



Question 78.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒



⇒


Question 79.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒



⇒


Question 80.

Mark (√) against the correct answer in each of the following:



A. tan (xex) + C

B. - tan (xex) + C

C. cot (xex) + C

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒ ⇒


⇒



Question 81.

Mark (√) against the correct answer in each of the following:



A. sec-1 x2 + C

B.

C. cosec-1 x2 + C

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒


⇒


⇒


Question 82.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


Put ⇒ x = t + 1 ⇒


⇒


⇒



Question 83.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,



Put ⇒


⇒


⇒



Question 84.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,



Put ⇒


⇒ ⇒ ⇒


⇒


Question 85.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,



Put ⇒


⇒


Put dt = 2z dz


⇒ ⇒


⇒


⇒


Question 86.

Mark (√) against the correct answer in each of the following:



A.

B.

C. log |sin x – cos x| + C

D. none of these
Answer:

Formula :-


Therefore ,


We can write




⇒


Put


⇒ ⇒


Question 87.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒


We can write




⇒


Put


⇒ ⇒


Question 88.

Mark (√) against the correct answer in each of the following:



A. log |sin x – cos x| + C

B.

C.

D.


Answer:

Formula :-


Therefore ,


⇒


We can write




⇒


Put


⇒ ⇒


Question 89.

Mark (√) against the correct answer in each of the following:



A. sin-1 (tan x) + C

B. cos-1 (sin x) + C

C. tan-1 (cos x) + C

D. tan-1 (sin x) + C
Answer:

Formula :-


Therefore ,


Put ⇒


⇒



Question 90.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒ ⇒


Put ⇒


⇒



Question 91.

Mark (√) against the correct answer in each of the following:



A.

B.

C. log|cos6 x| + C

D. none of these
Answer:

Formula :-


Therefore ,


⇒ ⇒


Put ⇒


⇒



Question 92.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 5 log |cos x| + C

D. none of these
Answer:

Formula :-


Therefore ,



Put ⇒


⇒



Question 93.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒


⇒




Put ⇒


⇒



Question 94.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒


⇒


Put ⇒


⇒



Question 95.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒



Put ⇒


⇒



Question 96.

Mark (√) against the correct answer in each of the following:



A. log {log (tan x )| + C

B.

C. log (sin x cos x) + C

D. none of these
Answer:

Formula :-


Therefore ,


⇒



Put ⇒ ⇒


⇒



Question 97.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒



Put ⇒


⇒


⇒



Question 98.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

Formula :-


Therefore ,


⇒ ⇒


Put ⇒


⇒ ⇒



Question 99.

Mark (√) against the correct answer in each of the following:



A. log |sin x – cos x| + C

B.

C. log |cos x + sin x| + C

D. none of these
Answer:

Formula :-


Therefore ,


⇒


Put ⇒


⇒ ⇒


Question 100.

Mark (√) against the correct answer in each of the following:



A.

B. e

C.

D. none of these
Answer:

Formula :-


Therefore ,



Put ⇒


⇒


Put dt = 2z dz


⇒ ⇒


⇒


⇒


Question 101.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

Let


Now multiplying and dividing by cos2x, we get,





Let tan x = t


Differentiating both sides, we get,


sec2x dx = dt


Therefore,



Integrating, we get,
Exercise 13b
Question 1.

Evaluate the following integrals:

(i)

(ii)


Answer:

i)


⇒


Now, we know that 1-cos2x=2sin2x


So, applying this identity in the given integral, we get,



⇒




Ans:


ii)


⇒


Now, we know that 1+cos2x=2cos2x


So, applying this identity in the given integral, we get,



⇒




Ans:



Question 2.

Evaluate the following integrals:

(i)

(ii)


Answer:

(i)


⇒


Now, we know that 1+cosx=2cos2 (x/2)


So, applying this identity in the given integral, we get,



⇒




Ans:


ii)


⇒


Now, we know that cosec2x-cot2 x=1


So, applying this identity in the given integral we get,


⇒


⇒


⇒


⇒-2cotx-x+c


⇒-2cotx-x+c


Ans: -2cotx-x+c
Question 3.

Evaluate the following integrals:

(i)

(ii)


Answer:

i)


⇒


Now, we know that 1-cos2nx=2sin2nx


So, applying this identity in the given integral, we get,



⇒




Ans:


(ii)


We know that 1-cos2x=sin2x


⇒


⇒Put cosx=t


⇒-sinxdx=dt


⇒


⇒


⇒


⇒


Resubstituting the value of t=cosx we get,


⇒


Ans:



Question 4.

Evaluate the following integrals:
Answer:

Substitute 3x+5=u


⇒3dx=du


⇒dx=du/3


⇒


Now We know that 1-cos2x=sin2x ,


⇒


⇒Substitute sinu=t


⇒cosu du=dt


⇒


⇒


⇒


⇒


Resubstituting the value of t=sinu and u=3x+5 we get,


⇒


Ans:



Question 5.

Evaluate the following integrals:
Answer:

⇒


Substitute 2x-3=u


⇒ 2dx=du


⇒dx=du/2


⇒


⇒ We know that 1-cos2x=sin2x


⇒


⇒Put cosu=t


⇒-sinxdu=dt


⇒


⇒


⇒


⇒


⇒


Resubstituting the value of t=cosu and u=2x-3 we get


⇒


⇒


Now as we know cos(-x)=cosx


⇒


=


Ans:



Question 6.

Evaluate the following integrals:

(i)

(ii)


Answer:

(i)


⇒


1-cos2x=2sin2x and 1+cos2x=2cos2x


⇒


⇒


Now sec2x-1=tan2x


⇒


⇒


⇒tanx-x+c


Ans: tanx-x+c


(ii)


⇒


1-cos2x=2sin2x and 1+cos2x=2cos2x


⇒


⇒


Now cosec2x-1=cot2x


⇒


⇒


⇒-cotx-x+c


Ans: -cotx-x+c
Question 7.

Evaluate the following integrals:

(i)

(ii)


Answer:

i)


⇒


1-cosx=2sin2x/2 and 1+cosx=2cos2x/2


⇒


⇒


Now sec2(x/2)-1=tan2(x/2)


⇒


⇒


⇒2tan(x/2)-x+c


Ans: 2tan(x/2)-x+c


(ii)


⇒


1-cosx=2sin2x/2 and 1+cosx=2cos2x/2


⇒


⇒


Now cosec2 (x/2)-1=cot2 (x/2)


⇒


⇒


⇒-2cot(x/2)-x+c


Ans: ⇒-2cot(x/2)-x+c
Question 8.

Evaluate the following integrals:
Answer:

⇒


Applying the formula: sinx×cosy=1/2(sin(x+y)-sin(y-x))


⇒


⇒


⇒


Ans:



Question 9.

Evaluate the following integrals:
Answer:

⇒


Applying the formula: cosx×cosy=1/2(cos(x+y)+cos(x-y))


⇒


⇒


⇒


Ans:



Question 10.

Evaluate the following integrals:
Answer:

⇒


Applying the formula: sinx×siny=1/2(cos(y-x)-cos(y+x))


⇒


⇒


⇒


Ans:



Question 11.

Evaluate the following integrals:
Answer:

⇒


Applying the formula: sinx×cosy=1/2(sin(y+x)-sin(y-x))


⇒


⇒


⇒


Ans:
Question 12.

Evaluate the following integrals:
Answer:

we know that 1+cos2x=2cos2x


So, applying this identity in the given integral we get,


⇒


⇒


⇒


Let sinx =t


⇒ cosx dx=dt


⇒


⇒


Resubstituting the value of t=sinx we get


⇒


Ans:
Question 13.

Evaluate the following integrals:
Answer:

⇒


⇒ …()


⇒


⇒


⇒


⇒ …(1+cos4x=2cos2x)


⇒


⇒


⇒


⇒


Ans:



Question 14.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒







Ans:



Question 15.

Evaluate the following integrals:
Answer:

Let sinx =t


⇒ cosx dx =dt


⇒


⇒


Resubstituting the value of t=sinx we get



Ans:



Question 16.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒Put tanx=t ⇒sec2dx=dt


⇒


⇒


Resubstituting the value of t=tanx we get


⇒


Ans:



Question 17.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒


Put sinx=t


⇒cosxdx=dt


⇒


⇒


⇒


Resubstituting the value of t=sinx we get,


⇒


Ans:



Question 18.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒


Put cosx=t


⇒-sinxdx=dt


⇒


⇒


⇒


Resubstituting the value of t=sinx we get,


⇒


Ans:



Question 19.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒


Put sinx=t


⇒cosxdx=dt


⇒


⇒



Resubstituting the value of t=sinx we get



Ans:



Question 20.

Evaluate the following integrals:
Answer:

⇒


⇒


⇒


Put cosx=t


⇒-sinxdx=dt


⇒


⇒



Resubstituting the value of t=cosx we get



Ans:



Question 21.

Evaluate the following integrals:
Answer:




cot2x=t ⇒-2cosec2 2xdx=dt





Resubstituting the value of t=cotx we get



Ans:



Question 22.

Evaluate the following integrals:
Answer:





Ans: 2sinx-log|secx+tanx|+c
Question 23.

Evaluate the following integrals:
Answer:




Now α is a constant



Ans:xcos α-sin αlog|cos(x+ α)|+c
Question 24.

Evaluate the following integrals:
Answer:




Now put cosx=t


⇒-sinxdx=dt




Resubstituting the value of t= cosx we get,



Ans:



Question 25.

Evaluate the following integrals:
Answer:




Put cosx =t


⇒ -sinxdx=dt



Now put t2-1=a


⇒2tdt=da


And t8=(a+1)4






Resubstituting the value of a=t2-1 and t=cosx ⇒a=cos2x-1=-sin2x we get





Ans:



Question 26.

Evaluate the following integrals:
Answer:

⇒


⇒ …()


⇒


⇒


⇒


⇒ …(1+cos8x=2cos2 4x)


⇒


⇒


⇒


⇒


Ans:



Question 27.

Evaluate the following integrals:
Answer:

Doing tangent half angle substitution we get,



Substitute u=tan(x/2)


⇒2du=sec2(x/2)dx


⇒dx=


⇒


⇒


⇒


⇒


Resubstituting the values we get,




Ans:



Question 28.

Evaluate the following integrals:
Answer:



Let tan=t


∴





Resubstituting the value of t we get



Ans:



Question 29.

Evaluate the following integrals:

a > 0 and b > 0
Answer:


Taking bcosx common from the denominator we get,




Let (a/b)+tanx=t


∴



Resubstituting the value of t = (a/b)+tanx we get



Ans:



Question 30.

Evaluate the following integrals:
Answer:



Let tan=t


∴



resubstituting the value of t we get



Ans:



Question 31.

Evaluate the following integrals:
Answer:






Ans: 4tanx-9cotx-25x+c
Question 32.

Evaluate the following integrals:
Answer:

⇒


Applying the formula: sinx×siny=1/2(cos(y-x)-cos(y+x))


⇒


⇒


⇒


⇒


Ans:



Question 33.

Evaluate the following integrals:
Answer:





Ans: -log(sinx+cosx)+c
Question 34.

Evaluate the following integrals:
Answer:



Let tan=t


∴





Resubstituting the value of t we get



Ans:



Exercise 13c
Question 1.

Evaluate the following integrals:
Answer:

Using BY PART METHOD.

Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x is the first function and is the second function.


Using Integration by part
Question 2.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x is the first function, and cos x is the second function.


Using Integration by part
Question 3.

Evaluate the following integrals:
Answer:

Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x is the first function and is the second function.


Using Integration by part
Question 4.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x is the first function, and Sin 3x is the second function.


Using Integration by part
Question 5.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x is the first function, and Cos 2x is the second function.


Using Integration by part
Question 6.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here log 2x is the first function, and x is the second function.


Using Integration by part
Question 7.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x is the first function, and cosec2x is the second function.


Using Integration by part
Question 8.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x2 is the first function, and cos x is the second function.


Using Integration by part




Again applying by the part method in the second half, we get
Question 9.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Using Integration by part



Writing Sin2x =


We have



Taking X as first function and Cos 2x as the second function.
Question 10.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Using Integration by part



Writing tan2x = sec2x - 1


We have



Using x as the first function and Sec2x as the second function
Question 11.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x2 is the first function, and ex is the second function.


Using Integration by part
Question 12.

Evaluate the following integrals:
Answer:

We know that Cos3x = 4Cos3x - 3Cosx

Cos3x =



Taking X2 as the first function and cos 3x and cos x as the second function and applying By part method.
Question 13.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here x2 is the first function, and e3x is the second function.


Using Integration by part
Question 14.

Evaluate the following integrals:
Answer:

We can write

We have



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x2 is the first function, and Cos 2x is the second function.


Using Integration by part
Question 15.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here log2x is the first function, and x3 is the second function.


Using Integration by part
Question 16.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here log(x + 1) is first function and x is second function.




Adding and subtracting 1 in the numerator,
Question 17.

Evaluate the following integrals:
Answer:

We can write it as

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here logx is the first function, and x - n is the second function.
Question 18.

Evaluate the following integrals:
Answer:

We can write it as

Let x2 = t


2xdx = dt


Using the relation in the above condition, we get



Integrating with respect to t


Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function, and et is the second function.




Replacing t with x2,we get
Question 19.

Evaluate the following integrals:
Answer:

We know that Sin3x = 3Sinx - 4Sin3x

Sin3x = (3Sinx - Sin3x)/4



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x is first function and sinx and sin3x as the second function.
Question 20.

Evaluate the following integrals:
Answer:

We can write cos3x = (cos3x + 3cosx)/4, we have


Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x is first function and cosx and cos3x as the second function.
Question 21.

Evaluate the following integrals:
Answer:

We can write it as


Now let x2 = t


2xdx = dt


Xdx = dt/2


Now



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and cost as the second function.




Replacing t with x2


=



Question 22.

Evaluate the following integrals:
Answer:

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here log(cosx) is the first function and sinx as the second function.
Question 23.

Evaluate the following integrals:
Answer:

We know that Sin2x = 2Sinxcosx


Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x is first function and sin2x as the second function.
Question 24.

Evaluate the following integrals:
Answer:

Let √x = t


We can write it as



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is first function and cos t as the second function.




Replacing t with √x


= 2√xsin√x + 2cos√x + c


= 2(cos√x + √xsin√x) + c
Question 25.

Evaluate the following integrals:
Answer:

We can write it as

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here cosecx is first function and cosec2x as the second function.




We know that Cot2x = Cosec2x - 1



We can write




Question 26.

Evaluate the following integrals:
Answer:

We can write it as

We also know that 2sinx.cosx = sin2x



We also know that



Here Sin4x = 2sin2x.cos2x



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here x is first function and Sin2x and sin4x as the second function.
Question 27.

Evaluate the following integrals:
Answer:

Let cosx = t

- sinxdx = dt


Now the integral we have is



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here logt is first function and 1 as the second function.




Replacing t with cosx
Question 28.

Evaluate the following integrals:
Answer:

Let logx = t

1/x dx = dt



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here logt is first function and 1 as the second function.




Now replacing t with logx
Question 29.

Evaluate the following integrals:
Answer:

=

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here log(2 + x2) is the first function and 1 as the second function.
Question 30.

Evaluate the following integrals:
Answer:

We can write it as

Using by part and ILATE


Taking x as first function and sec2x and secxtanx as the second function, we have
Question 31.

Evaluate the following integrals:
Answer:

Let us assume logx = t

X = et


dx = etdt


Now we have



Considering f(x) = 1/t ; f’(x) = - 1/t2



By the integral property of


So the solution of the integral is



Substituting the value of t as logx
Question 32.

Evaluate the following integrals:
Answer:

We know that

Putting in the original equation



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here cos6x and cos2x is first function and e - x as the second function.



Solving both parts individually



Solving the second part,



Putting in the obtained equation
Question 33.

Evaluate the following integrals:
Answer:

Let √x = t


Replacing in the original equation , we get



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and et as the second function.




Replacing t with √x


= 2e√x(√x - 1) + c
Question 34.

Evaluate the following integrals:
Answer:

We can write Sin2x = 2sinx.cosx


Let Sinx = t


Cosxdx = dt



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and et as the second function.




Replacing t with sin x


= 2esinx(sinx - 1) + c
Question 35.

Evaluate the following integrals:
Answer:

Let sin - 1x = t

X = sint



Putting this in the original equation, we get



Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and sin t as the second function.




We can write cos t = √1 - sin2t


= - t(√1 - sin2t) + sint + c


Now replacing sin - 1x = t


= - sin - 1x(√1 - x2) + x + c
Question 36.

Evaluate the following integrals:
Answer:

Let tan - 1 x = t and x = tan t

Differentiating both sides, we get



Now we have




Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and sec2t as the second function.




We know that sec t = √tan2t + 1
Question 37.

Evaluate the following integrals:
Answer:

We can write it as

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here log(x + 2) is first function and (x + 2) - 2 as second function.
Question 38.

Evaluate the following integrals:
Answer:

Let x = sin t ; t = sin - 1x

dx = cos t dt



We know that sin 2t = 2 sint×cost


We have


Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here t is the first function and sin 2t as the second function.




We know that cos2t = 1 - 2sin2t , sin2t = 2sint×cost and cos t = √1 - sin2t


Replacing in above equation
Question 39.

Evaluate the following integrals:
Answer:

Let x = cos t ; t = cos - 1x

dx = - sin t dt



We know that sin 2t = 2 sint×cost


We have


Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking first function to the one which comes first in the list.


Here t is first function and sin 2t as second function.




We know that cos2t = 2cos2t - 1 and sin2t = 2sint×cost and sint = √1 - cos2t


Replacing in above equation
Question 40.

Evaluate the following integrals:
Answer:

We can write it as

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.


Here cot - 1x is first function and 1 as the second function.




Let 1 + x2 = t


2xdx = dt


Xdx = dt/2



Now replacing t with 1 + x2


= xcot - 1x + log(1 + x2)/2 + c
Question 41.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cot-1x and f2(x) = x,








, where c is the integrating constant
Question 42.

Evaluate the following integrals:



[CBSE 2006C]
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cot-1x and f2(x) = x2,






Taking (1+x2)=a,


2xdx=da i.e. xdx=da/2


Again, x2=a-1






Replacing the value of a, we get,






The total integration yields as


, where c is the integrating constant
Question 43.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin-1√x and f2(x) = 1,






Taking (1-x)=a2,


-dx=2ada i.e. dx=-2ada


Again, x=1-a2






Replacing the value of a, we get,




The total integration yields as


, where c is the integrating constant
Question 44.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cos-1√x and f2(x) = 1,






Taking (1-x)=a2,


-dx=2ada i.e. dx=-2ada


Again, x=1-a2






Replacing the value of a, we get,




The total integration yields as


, where c is the integrating constant
Question 45.

Evaluate the following integrals:
Answer:

Formula to be used – We know , cos3x = 4cos3x-3cosx



Assuming x = cosa, 4cos3a-3cosa=cos3a


And, dx = -sinada


Hence, a=cos-1x


Again, sina=√(1-x2)





Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = a and f2(x) = sina,






Replacing the value of a we get,



, where c is the integrating constant
Question 46.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = and f2(x) = 1,







Now,





Again, we know,




Replacing x by tanx, it is obtained that,



So, the final integral yielded is


, where c is the integrating constant
Question 47.

Evaluate the following integrals:
Answer:

Formula to be used – We know,



Assuming x = tana,



And, dx = sec2ada


Hence, a=tan-1x


Now, sec2a-tan2a=1 , so,seca=√(1+x2)





Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = a and f2(x) = sec2a,






Replacing the value of a we get,



, where c is the integrating constant
Question 48.

Evaluate the following integrals:
Answer:

Formula to be used – We know,



Assuming x = tana,



And, dx = sec2ada


Hence, a=tan-1x


Now, sec2a-tan2a=1 , so, seca=√(1+x2)





Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = a and f2(x) = sec2a,






Replacing the value of a we get,



, where c is the integrating constant
Question 49.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin-1x and f2(x) = 1/x2,






Taking x= sina, dx = cosada


Hence, coseca=1/x


Now, cosec2a-cot2a = 1 so cota=√(1-x2)/x






Replacing the value of a, we get,




The total integration yields as


, where c is the integrating constant
Question 50.

Evaluate the following integrals:
Answer:

Say, tanx = a


Hence, sec2xdx=da




Now, taking 1-a2 = k , -2ada=dk i.e. ada=-dk/2





Replacing the value of k,




Replacing the value of a,



, where c is the integrating constant
Question 51.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin4x and f2(x) = e3x,











, where c is the integrating constant
Question 52.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sinx and f2(x) = e2x,











, where c is the integrating constant
Question 53.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin2x and f2(x) = e2x,














, where c is the integrating constant
Question 54.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cos(3x+4) and f2(x) = e2x,











, where c is the integrating constant
Question 55.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cosx and f2(x) = e-x,










, where c is the integrating constant
Question 56.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sinx and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 57.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cotx and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 58.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = secx and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 59.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = tan-1x and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 60.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = logsinx and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 61.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = logcosx and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 62.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = logcosx and f2(x) = ex in the second integral and keeping the first integral intact,






, where c is the integrating constant
Question 63.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = tanx and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 64.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cotx and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 65.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = secx and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 66.

Evaluate the following integrals:
Answer:





Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cotx and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 67.

Evaluate the following integrals:
Answer:









Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = tan(x/2) and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 68.

Evaluate the following integrals:
Answer:





Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = cot2x and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 69.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin-1x and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 70.

Evaluate the following integrals:
Answer:




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = logx and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 71.

Evaluate the following integrals:
Answer:



For x=-1, equation: -1 = B i.e. B = -1


For x=0, equation: 0 = A-1 i.e. A = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1+x) and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 72.

Evaluate the following integrals:
Answer:



For x=-1, equation: -2 = C i.e. C = -2


For x=0, equation: -1 = A+B-2 i.e. A+B = 1


For x=1, equation: 0 = 4A+2B-2


i.e. 2(A+B+A) = 2


⇨1+A = 1


⇨A = 0


And, B = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1+x)2 and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 73.

Evaluate the following integrals:
Answer:



For x=1, equation: 1 = B i.e. B = 1


For x=2, equation: 0 = -A+1 i.e. A = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1-x) and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 74.

Evaluate the following integrals:
Answer:



For x=1, equation: -2 = C i.e. C = -2


For x=0, equation: -3 = A-B-2 i.e. B = A+1


For x=3, equation: 0 = 4A+2B-2


i.e. 2(A+B+A) = 2


⇨1+3A = 1


⇨A = 0


And, B = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1-x)2 and f2(x) = ex in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 75.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/3xand f2(x) = e3x in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 76.

Evaluate the following integrals:
Answer:



For x=-2, equation: -1 = B i.e. B = -1


For x=-1, equation: 0 = A-1 i.e. A = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(x+2) and f2(x) = ex in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 77.

Evaluate the following integrals:
Answer:



For x=-1/2, equation: -1/2 = B i.e. B = -1/2


For x=0, equation: 0 = A-1/2 i.e. A = 1/2




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1+2x) and f2(x) = e2x in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 78.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/2x and f2(x) = e2x in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 79.

Evaluate the following integrals:
Answer:



Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = logx and f2(x) = ex in the first integral and keeping the second integral intact,







, where c is the integrating constant
Question 80.

Evaluate the following integrals:
Answer:



For x=1, equation: 0 = A+B


For x=1/e, equation: -1 = B i.e. B = -1


So, A = 1




The given equation becomes




Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(1+logx) and f2(x) = 1in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 81.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = sin(logx) and f2(x) = 1in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 82.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = 1/(logx) and f2(x) = 1in the first integral and keeping the second integral intact,





, where c is the integrating constant
Question 83.

Evaluate the following integrals:
Answer:

Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = log(logx) and f2(x) = 1in the first integral and keeping the second integral intact,







, where c is the integrating constant
Question 84.

Evaluate the following integrals:
Answer:

It is know that sin-1x+cos-1x = π/2




Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Now, for the first term,


Taking f1(x) = sin-1√x and f2(x) = 1,






Taking (1-x)=a2,


-dx=2ada i.e. dx=-2ada


Again, x=1-a2






Replacing the value of a, we get,




The total integration yields as


, where c’ is the integrating constant


For the second term,


Taking f1(x) = cos-1√x and f2(x) = 1,






Taking (1-x)=a2,


-dx=2ada i.e. dx=-2ada


Again, x=1-a2






Replacing the value of a, we get,




The total integration yields as


, where c’’ is the integrating constant





where c is the integrating constant
Question 85.

Evaluate the following integrals:
Answer:

Tip – 5x is to be replaced by a





The equation becomes as follows:



Tip – 5a is to be replaced by k





The equation becomes as follows:





Re-replacing the value of k,



Re-replacing the value of a,


, where c is the integrating constant
Question 86.

Evaluate the following integrals:
Answer:









Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = tanx and f2(x) = e2x in the second integral and keeping the first integral intact,





, where c is the integrating constant
Question 87.

Evaluate the following integrals:
Answer:









Tip – If f1(x) and f2(x) are two functions, then an integral of the form can be INTEGRATED BY PARTS as


where f1(x) and f2(x) are the first and second functions respectively.


Taking f1(x) = tanx and f2(x) = e2x in the second integral and keeping the first integral intact,





, where c is the integrating constant
Objective Questions Ii
Question 1.

Mark (√) against the correct answer in each of the following:



A. ex (1 – x) + C

B. ex (x – 1) + C

C. ex (x – 1) + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)







Ans ) c
Question 2.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)










Ans ) B
Question 3.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 2x sin 2x + 4 cos 2x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let 2x = t







Taking 1st function as and second function as









Question 4.

Mark (√) against the correct answer in each of the following:



A. x tan x – log |cos x| + C

B. x tan x + log |cos x| + C

C. x tan x + log |sec x| + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as








Question 5.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let 2x = t







Taking 1st function as and second function as









Question 6.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as









Question 7.

Mark (√) against the correct answer in each of the following:



A. x cot x – log |sin x| + C

B. - cot x + log |sin x| + C

C. x tan x – log |sec x| + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)






Ans ) D None of these
Question 8.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

To find: Value of


Formula used:



We have, … (i)
Question 9.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 10.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 11.

Mark (√) against the correct answer in each of the following:



A.

B.

C. x (log x + 1) + C

D. x (log x – 1) + C
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as









Question 12.

Mark (√) against the correct answer in each of the following:



A.

B.

C. x (log x – 1) loge 10 + C

D. x(log x – 1) log10 e + C
Answer:

To find: Value of


Formula used:


We have, … (i)




Taking 1st function as and second function as









Question 13.

Mark (√) against the correct answer in each of the following:



A.

B.

C. x (log x)2 – 2x log x + 2x + C

D. x (log x)2 + 2x log x – 2x + C
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as









Question 14.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:


We have, … (i)


Putting




⇒ dx = 2t dt
Question 15.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Putting




⇒ dx = 2t dt
Question 16.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 2x [cos (log x) + sin (log x)] + C

D. 2x [cos (log x) – sin (log x)] + C
Answer:

To find: Value of


Formula used:



We have, … (i)



Taking cos(logx) as first function and 1 as second function.
Question 17.

Mark (√) against the correct answer in each of the following:



A.

B.

C. 2{sec x tan x + log |sec x + tan x|}+C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)



Taking secx as first function and sec2x as second function.
Question 18.

Mark (√) against the correct answer in each of the following:



A. x log x + C

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 19.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 20.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 21.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 22.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:


We have, … (i)
Question 23.

Mark (√) against the correct answer in each of the following:



A. (2 sin x) esin x + C

B. (2 cos x) esin x + C

C. 2esin x (sin x + 1) + C

D. 2esin x (sin x – 1) + C
Answer:

To find: Value of


Formula used:



We have, … (i)



Put sinx = t
Question 24.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)




Putting sin-1x = t , x = sint
Question 25.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:


We have, … (i)



Putting tan-1x = t , x = tant


dx = sec2t dt


When x = tant













Taking 1st function as and second function as









Question 26.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as










Question 27.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let,



⇒ dx = 2t dt





Taking 1st function as and second function as











Question 28.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let, ⇒ x = cosθ


⇒ dx = -sinθ dθ


If x = cosθ ,


Then = sinθ




Taking 1st function as and second function as









Question 29.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let, ⇒ x = tanθ


⇒ dx = sec2θ dθ


If x = tanθ ,


Then 1 + x2 = sec2θ


⇒ θ = sec-1




Taking 1st function as and second function as









Question 30.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let, ⇒ x = secθ


⇒ dx = secθ tanθ dθ


If x = secθ ,


Then = tanθ




Taking 1st function as and second function as








Question 31.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let x = sinθ , ⇒ θ = sin-1x


⇒ dx = cosθ dθ


If x = sinθ ,


Then = cosθ







Taking 1st function as and second function as










Question 32.

Mark (√) against the correct answer in each of the following:



A. 2x tan-1 x + log |1 + x2| + C

B. 2x tan-1 x - log |1 + x2| + C

C. 2x sin-1 x + log |1 + x2| + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let x = tanθ , ⇒ θ = tan-1x


⇒ dx = sec2θ dθ


If x = tanθ ,


Then 1 + x2 = sec2θ


⇒ θ = sec-1







Taking 1st function as and second function as










Question 33.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:


We have, … (i)


Let x = cosθ , ⇒ θ = cos-1x


⇒ dx = -sinθ dθ


If x = cosθ ,


Then = sinθ









Taking 1st function as and second function as









Question 34.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

To find: Value of


Formula used:


We have, … (i)


Let x = tanθ , ⇒ θ = tan-1x


⇒ dx = sec2θ dθ


If x = tanθ ,


Then 1 + x2 = sec2θ


⇒ θ = sec-1







Taking 1st function as and second function as










Question 35.

Mark (√) against the correct answer in each of the following:



A. x2 sin x + 2x cos x – 2 sin x + C

B. 2x cos x – x sin x + 2 sin x + C

C. x2 sin x – 2x sin x + 2 sin x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as





Taking 1st function as and second function as








Question 36.

Mark (√) against the correct answer in each of the following:



A. cos x log (cos x) – cos x + C

B. -cos x log (cos x) + cos x + C

C. cos x log (cos x) + cos x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let cosx = t


-sinx dx = dt





Taking 1st function as and second function as 1
Question 37.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:


We have, … (i)




Let 2x = t


2dx = dt





Taking 1st function as and second function as sint
Question 38.

Mark (√) against the correct answer in each of the following:



A. x2 sin x2 + cos x2 + C

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let x2 = t


⇒ xdx = dt






Taking 1st function as and second function as








Question 39.

Mark (√) against the correct answer in each of the following:



A. 2x tan-1 x + log(1 + x2) + C

B. -2x tan-1 x – 2 log (1 + x2) + C

C. 2x tan-1 x – log (1 + x2) + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Let x = tant , t = tan-1x


⇒ dx = sec2t dt


If tant = x ,


sec t = 1 + x 2







Taking 1st function as and second function as








Question 40.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)


Taking 1st function as and second function as











Question 41.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)



Taking 1st function as sin(logx) and second function as 1





Taking 1st function as cos(logx) and second function as 1
Question 42.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D.


Answer:

To find: Value of


Formula used:



We have, … (i)


Putting sint = x, ⇒ t =


⇒ dx = cost dt


When x = sint then





Taking 1st function as t2 and second function as cost





Taking 1st function as t and second function as sint
Question 43.

Mark (√) against the correct answer in each of the following:



A.

B. xex – ex + C

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 44.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 45.

Mark (√) against the correct answer in each of the following:



A.

B. ex sin-1 x + C

C.

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 46.

Mark (√) against the correct answer in each of the following:



A. ex log sec x + C

B. ex tan x + C

C. ex (log cos x) + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 47.

Mark (√) against the correct answer in each of the following:



A. ex log sec x + C

B. ex tan x + C

C. ex (log cos x) + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 48.

Mark (√) against the correct answer in each of the following:



A. ex log (sec x + tan x) + C

B. ex sec x + C

C. ex log tan x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 49.

Mark (√) against the correct answer in each of the following:



A.

B. ex tan-1 x + C

C. -ex cot-1 x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 50.

Mark (√) against the correct answer in each of the following:



A. ex tan x + C

B. ex log cos x + C

C. ex log sec x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 51.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. None of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 52.

Mark (√) against the correct answer in each of the following:



A. ex sin x + C

B. ex cos x + C

C. ex tan x + C

D. None of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 53.

Mark (√) against the correct answer in each of the following:



A. ex (1 + tan x) + C

B. ex sec x + C

C. ex tan x + C

D. none of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 54.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. None of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 55.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. None of these
Answer:

To find: Value of


Formula used:



We have, … (i)
Question 56.

Mark (√) against the correct answer in each of the following:



A.

B.

C.

D. None of these
Answer:

To find: Value of


Formula used:



We have, … (i)