Indefinite Integral
Class 12th Mathematics RS Aggarwal Solution
- Evaluatei. integrate x^{7} dx ii. integrate x^{-7} dx iii. integrate…
- i. integrate ( 6x^{5} - {2}/{ x^{4} } - 7x + frac {3}/{x} - 5+4e^{x} +…
- i. integrate (2-5x) (3+2x) (1-x) dx ii. integrate root {x} ( ax^{2} + bx+c…
- i. integrate ( x^{2} - {1}/{ x^{2} } ) ^{3} dx ii. integrate ( root {x} -…
- integrate [1 + {1}/{ ( 1+x^{2} ) } - frac {2}/{ root { 1-x^{2} } } + frac…
- i. integrate ( { x^{2} - 1 }/{ x^{2} + 1 } ) dx ii. integrate ( { x^{6}…
- integrate ( 9sinx-7cosx - {6}/{cos^{2}x} + frac {2}/{sin^{2}x}+cot^{2}x ) dx…
- integrate ( {cotx}/{sinx}-tan^{2}x - frac {tanx}/{cosx} + frac…
- i. integrate secx (secx+tanx) dx ii. integrate cosecx (cosecx-cotx) dx…
- i. integrate (tanx+cotx)^{2} dx ii. integrate ( {1+2sinx}/{cos^{2}x} )…
- i. integrate {1}/{ (1-cosx) } dx ii. integrate {1}/{ (1-sinx) } dx…
- i. integrate {tanx}/{ (secx+tanx) } dx ii. integrate {cosecx}/{…
- i. integrate {cosx}/{1+cosx}dx ii. integrate {sinx}/{ (1-sinx) } dx…
- i. integrate root {1+cos2x}dx ii. integrate root {1-cos2x}dx Evaluate :…
- i. integrate {1}/{ (1+cos2x) } dx ii. integrate {1}/{ (1-cos2x) } dx…
- integrate root {1+sin2x}dx Evaluate :
- integrate ( {sin^{3}x+cos^{3}x}/{sin^{2}xcos^{2}x} ) dx Ans. sec x – cosec…
- integrate tan^{-1} ( {sin2x}/{1+cos2x} ) dx Ans. { x^{2} }/{2}+c…
- integrate cos^{1} ( {1-tan^{2}x}/{1+tan^{2}x} ) dx Ans. x^{2} + c Evaluate…
- integrate cos^{-1} (sinx) dx Ans. ( { pi x }/{2} - frac { x^{2} }/{2}+c )…
- integrate tan^{-1}root { {1-sinx}/{1+sinx} } dx Ans. { pi x }/{4} -…
- integrate (3cotx-2tanx)^{2} dx Ans. 4 tan x – 9 cos x – 25x + C Evaluate :…
- integrate (3sinx+4cosecx)^{2} dx Ans. {57}/{2} x - frac {9}/{4}…
- integrate {dx}/{ ( root {x+1} + sqrt{x+2} ) } Ans. {2}/{3} (x+2)^{3/2}…
- integrate {dx}/{ ( root {x+3} - sqrt{x+2} ) } Ans. {2}/{3} (x+3)^{3/2}…
- integrate ( {1+cosx}/{1-cosx} ) dx Ans. -2cot {x}/{2} - x+c Evaluate :…
- integrate { (1+tanx) }/{ (1-tanx) } dx Ans. -log |cos x – sin x| + C…
- integrate { cos (x+a) }/{ sin (x+b) } dx Ans. cos(a – b)log |sin (x + b)|…
- integrate { sin ( x - alpha ) }/{ sin ( x + alpha ) } dx Ans. x cos 2α -…
- integrate (1-x) root {x}dx Ans. {2}/{15} x root {x} (5-3x) + c Evaluate :…
- integrate {sec^{2}x}/{cosec^{2}x}dx Ans. tan x – x + C Evaluate :…
- integrate { {2-3sinx}/{cos^{2}x} } dx Ans. 2 tan x – 3 sec x + C Evaluate :…
- integrate x^{6} dx = ? Mark (√) against the correct answer in each of the following:…
- integrate x^{5/3}dx = ? Mark (√) against the correct answer in each of the following:…
- integrate {1}/{ x^{3} } dx = ? Mark (√) against the correct answer in each of the…
- integrate cube root {x}dx = ? Mark (√) against the correct answer in each of the…
- integrate {1}/{ cube root {x} } dx = ? Mark (√) against the correct answer in each…
- integrate cube root { x^{2} } dx = ? Mark (√) against the correct answer in each of…
- integrate 3^{x} dx = ? Mark (√) against the correct answer in each of the following:…
- integrate 2^{logx}dx = ? Mark (√) against the correct answer in each of the following:…
- integrate cosecx (cosecx+cotx) dx = ? Mark (√) against the correct answer in each of…
- integrate {secx}/{ (secx+tanx) } dx = ? Mark (√) against the correct answer in each…
- integrate { (1-cos2x) }/{ (1+cos2x) } dx = ? Mark (√) against the correct answer in…
- integrate {1}/{sin^{2}xcos^{2}x}dx = ? Mark (√) against the correct answer in each…
- integrate {cos2x}/{cos^{2}xsin^{2}x}dx = ? Mark (√) against the correct answer in…
- integrate { (cos2x-cos2alpha ) }/{ (cosx-cosalpha) } dx = ? Mark (√) against the…
- integrate root {1+cos2x}dx = ? Mark (√) against the correct answer in each of the…
- integrate root {1+sin2x}dx = ? Mark (√) against the correct answer in each of the…
- integrate {cos2x}/{sin^{2}xcos^{2}x}dx = ? Mark (√) against the correct answer in…
- integrate {dx}/{ (1-cos2x) } = ? Mark (√) against the correct answer in each of the…
- integrate {sin2x}/{sinx}dx = ? Mark (√) against the correct answer in each of the…
- integrate { (1-sinx) }/{cos^{2}x}dx = ? Mark (√) against the correct answer in each…
- integrate cot^{2}xdx = ? Mark (√) against the correct answer in each of the…
- integrate secx (secx+tanx) dx = ? Mark (√) against the correct answer in each of the…
- integrate {sec^{2}x}/{cosec^{2}x}dx = ? Mark (√) against the correct answer in each…
- integrate {sin^{2}x}/{ (1+cosx) } dx = ? Mark (√) against the correct answer in…
- integrate {cotx}/{ (cosecx-cotx) } dx = ? Mark (√) against the correct answer in…
- integrate {sinx}/{ (1+sinx) } dx = ? Mark (√) against the correct answer in each of…
- integrate { (1+sinx) }/{ (1-sinx) } dx = ? Mark (√) against the correct answer in…
- integrate {1}/{ (1+cosx) } dx = ? Mark (√) against the correct answer in each of…
- integrate sin^{-1} (cosx) dx = ? Mark (√) against the correct answer in each of the…
- integrate tan^{-1} { root { {1-cos2x}/{1+cos2x} } } dx = ? Mark (√) against the…
- integrate cot^{-1} ( {sin2x}/{1-cos2x} ) dx = ? Mark (√) against the correct answer…
- integrate sin^{-1} ( {2tanx}/{1+tan^{2}x} ) dx = ? Mark (√) against the correct…
- integrate cos^{-1} ( {1-tan^{2}x}/{1+tan^{2}x} ) dx = ? Mark (√) against the…
- integrate tan^{-1} (cosecx-cotx) dx = ? Mark (√) against the correct answer in each…
- integrate ( { ( x^{4} + 1 ) }/{ ( x^{2} + 1 ) } ) dx = ? Mark (√) against the…
- integrate { (ax+b) }/{ (cx+d) } dx = ? Mark (√) against the correct answer in each…
- integrate { (sin^{3}x+cos^{3}x) }/{sin^{2}xcos^{2}x}dx = ? Mark (√) against the…
- integrate {sinx}/{ sin ( x - alpha ) } dx = ? Mark (√) against the correct answer…
- integrate sin3xsin2xdx = ? Mark (√) against the correct answer in each of the…
- integrate cos3xsin2xdx = ? Mark (√) against the correct answer in each of the…
- integrate cos4xcosxdx = ? Mark (√) against the correct answer in each of the…
Exercise 12
Question 1.Evaluate :
Evaluate
i. 
ii. 
iii. 
iv. 
v. 
vi. 
vii. 
viii. 
ix. 
Answer:
i Given:
ii. Given:
iii. Given:
iv. Given:
v. Given:
vi. Given:
vii. Given:
viii. Given:
ix. Given:
Question 2.
Evaluate :
i. 
ii. 
iii. 
Answer:
i Given:
ii. Given:
iii. Given:
Question 3.
Evaluate :
i. 
ii. 
iii. 
Answer:
i. Given:
∫(2-5x)(3+2x)(1-x)dx
=∫(6-11x-10x2 )(1-x) dx
=∫(10 x3+x2-17x+6)dx
ii. Given:
iii. Given:
Question 4.
Evaluate :
i. 
ii. 
iii. 
iv. 
v. 
vi. 
Answer:
i. Given:
=∫x6 +x-6 -3x2-3x-2 dx
ii. Given:
iii. Given:
iv. Given:
v. Given:
vi. Given:
=x2- 4x+8x+8 ln|x-2|+x + c
=x2 +5x+8 ln|x-2|+c
Question 5.
Evaluate :
Answer:
Given:
Since, 
;
;
&
So,
Question 6.
Evaluate :
i. 
ii. 
iii. 
iv. 
Answer:
i. Given:
=x-2 tan-1 x + c
ii. Given:
iii. Given:
iv. Given:
=x - tan-1 x + c
Question 7.
Evaluate :
Answer:
Given:
=-9 cos x-7 sin x-6 tan x-3 cot x – x + c
Question 8.
Evaluate :
Ans. - cosec x + tan x + x – sec x + C
Answer:
Given:
=-csc x-tan x + x-sec x+2 tan x + c
=-csc x + tan x + x -sec x + c
Question 9.
Evaluate :
i. 
ii. 
Answer:
i. Given:
=tan x+sec x+c
ii. Given:
=-cot x + csc x + c
Question 10.
Evaluate :
i. 
ii. 
iii. 
Answer:
i. Given:
=tan x-cot x+c
ii. Given:
=tan x+2 sec x + c
iii. Given:
=-2 csc x-4 cot x + c
Question 11.
Evaluate :
i. 
ii. 
Answer:
i. Given:
Multiply and divide by 
=-cot x-csc x + c
ii. Given:
Multiply and divide by 
=tan x + sec x + c
Question 12.
Evaluate :
i. 
ii. 
Answer:
i. Given:
Multiply and divide by 
=sec x-tan x + x + c
ii. Given:
Multiply and divide by 
=-cot x-csc x + c
Question 13.
Evaluate :
i. 
ii. 
Answer:
i. Given:
Multiply and divide by 
=-csc x + cot x + x + c
ii. Given:
Multiply and divide by
=sec x + tan x-x + c
Question 14.
Evaluate :
i. 
ii. 
Answer:
i. Given:
ii. Given:
=-√2 cos x + c
Question 15.
Evaluate :
i. 
ii. 
Answer:
i. Given:
ii. Given:
Question 16.
Evaluate :
Answer:
Given:
=∫(cos x + sin x) d
=sin x-cos x+c
Question 17.
Evaluate :
Ans. sec x – cosec x + C
Answer:
Given:
=∫(tan x sec x + csc x cot x )dx
=sec x- csc x + c
Question 18.
Evaluate :
Ans. 
Answer:
Given:
=∫tan-1 (tan x )dx
=∫x dx
Question 19.
Evaluate :
Ans. 
Answer:
Given:
=∫cos-1(cos 2x )dx
=∫2x dx
=x2 +c
Question 20.
Evaluate :
Ans. 
Answer:
Given:
Question 21.
Evaluate :
Ans. 
Answer:
Given:
Question 22.
Evaluate :
Ans. 4 tan x – 9 cos x – 25x + C
Answer:
Given:
=-9 cot x+4 tan x-25x+c
Question 23.
Evaluate :
Ans. 
Answer:
Given:
Question 24.
Evaluate :
Ans. 
Answer:
Given:
Multiply and divide by 
Question 25.
Evaluate :
Ans. 
Answer:
Given:
Multiply and divide by 
Question 26.
Evaluate :
Ans. 
Answer:
Given:
Multiply and divide by
Question 27.
Evaluate :
Ans. -log |cos x – sin x| + C
Answer:
Given:
Let cos x-sin x=t
(-sin x-cos x) dx=dt
-(sin x + cos x )dx=dt
dx=-dt
So, I
=-ln|t|+c
=-ln|cos x-sin x |+c
Question 28.
Evaluate :
Ans. cos(a – b)log |sin (x + b)| -x sin(a –b) + C
Answer:
Given:
=cos(a-b)∫cot(x + b) dx-∫sin(a-b)dx
=cos (a-b)ln|sin(x + b)|-x sin(a-b)+c
Question 29.
Evaluate :
Ans. x cos 2α - sin 2α . log |sin(x + α)| + C
Answer:
Given:
=∫cos 2α dx-sin2α ∫cot(x+α)dx
=x cos 2α-sin 2α×ln|sin(x+α) |+c
Question 30.
Evaluate :
Ans. 
Answer:
Given:
Question 31.
Evaluate :
Ans. tan x – x + C
Answer:
Given:
=∫(tan x )2 dx
=∫((sec x )2-1)dx
=tan x – x + c
Question 32.
Evaluate :
Ans. 2 tan x – 3 sec x + C
Answer:
Given:
=2 tan x-3 sec x+c
Objective Questions
Question 1.Mark (√) against the correct answer in each of the following:
A. 7x7 + C
B. 
C. 
D. 
Answer:
Given:
∫x6 dx,
Question 2.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 3.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 4.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 5.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 6.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 7.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 8.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
2adc
D. 
Answer:
Given:
As 
Question 9.
Mark (√) against the correct answer in each of the following:
A. cot x – cosec x + C
B. - cot x + cosec x + C
C. cot x + cosec x + C
D. - cot x – cosec x + C
Answer:
Given:
=-cot x-csc x+c
Question 10.
Mark (√) against the correct answer in each of the following:
A. tan x + sec x + C
B. tan x – sec x + C
C. - tan x + sec x + C
D. - tan x – sec x + C
Answer:
Given:
Multiply and divide by (sec x-tan x )
=∫(sec x )2 -tan x sec x dx
=tan x-sec x + c
Question 11.
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. - tan x – x + C
Answer:
Given:
=∫(tan x )2
=∫((sec x )2 -1) dx
=tan x-x+c
Question 12.
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:
As we know sin2x + cos2 x = 1
=∫(sec x )2+(csc x )2 dx
=tan x-cot x+c
Question 13.
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:
=∫(csc x )2-(sec x )2 dx
=-cot x-tan x+c
Question 14.
Mark (√) against the correct answer in each of the following:
A. 2 sin x + 2x cos α + C
B. 2 sin x – 2x cos α + C
C. - 2 sin x + 2x cos α + C
D. - 2 sin x – 2x cos α + C
Answer:
Given:
=2∫cos (x)+cos(α)dx
=2 sin x+2x cos α+c
Question 15.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
=√2 ∫cos x dx
=√2 sin x+c
Question 16.
Mark (√) against the correct answer in each of the following:
A. sin x + cos x + C
B. - sin x + cos x + C
C. sin x – cos x + C
D. - sin x – cos x + C
Answer:
Given:
=∫cos x + sin x dx
=sin x-cos x+c
Question 17.
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:
=∫(csc x )2-(sec x )2 dx
=-cot x-tan x+c
Question 18.
Mark (√) against the correct answer in each of the following:
A. 
B. 2 cot x + C
C. 
D. -2 cot x + C
Answer:
Given:
Question 19.
Mark (√) against the correct answer in each of the following:
A. 2 sin x + C
B. 
C. 2 cos x + C
D. 
Answer:
Given:
=2∫cos x dx
=2 sin x+c
Question 20.
Mark (√) against the correct answer in each of the following:
A. tan x + sec x + C
B. tan x – sec x + C
C. - tan x + sec x + C
D. - tan x – sec x + C
Answer:
Given:
=∫(sec x )2-tan x sec x dx
=tan x-sec x+c
Question 21.
Mark (√) against the correct answer in each of the following:
A. - cot x – x + C
B. cot x – x + C
C. - cot x + x + C
D. cot x + x + C
Answer:
Given:
∫cot2 x dx=∫((csc x )2-1)dx
=-cot x-x+c
Question 22.
Mark (√) against the correct answer in each of the following:
A. tan x - sec x + C
B. - tan x + sec x + C
C. tan x + sec x + C
D. - tan x – sec x + C
Answer:
Given:
∫sec x (sec x + tan x) dx= ∫(sec x )2+sec x tan x dx
=tan x+ sec x + c
Question 23.
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. - tan x - x + C
Answer:
Given:
=∫(tan x )2 dx
=∫(sec x )2-1 dx
=tan x-x+c
Question 24.
Mark (√) against the correct answer in each of the following:
A. x + sin x + C
B. x – sin x + C
C. sin x – x + C
D. - sin x – x + C
Answer:
Given:
=∫(1-cos x )dx
=x-sin x+c
Question 25.
Mark (√) against the correct answer in each of the following:
A. - cosec x – cot x – x + C
B. cosec x – cot x – x + C
C. - cosec x + cot x – x + C
D. cosec x + cot x – x + C
Answer:
Given:
=∫cot x csc x+(csc x )2 dx
=-csc x-cot x+c
Question 26.
Mark (√) against the correct answer in each of the following:
A. sec x + tan x + x + C
B. sec x – tan x + x + C
C. - sec x + tan x + x + C
D. None of these
Answer:
Given:
Multiply and divide by (1-sin x )
=∫(tan x sec x-(tan x )2 )dx
=∫(tan x sec x-(sec x )2+1)dx
=sec x-tan x+x+c
Question 27.
Mark (√) against the correct answer in each of the following:
A. 2 tan x + 2 sec x + x + C
B. 2 tan x + 2 sec x – x + C
C. tan x + sec x – x + C
D. None of these
Answer:
Given:
Multiply and divide with (1 + sin x ) to get,
=∫(sec x )2+(tan x )2+2 tan x sec x dx
=∫2(sec x )2-1+2 tan x sec x dx
=2 tan x-x+2 sec x+c
Question 28.
Mark (√) against the correct answer in each of the following:
A. - cot x + cosec x + C
B. cot x – cosec x + C
C. cot x + cosec x + C
D. None of these
Answer:
Given:
Multiply and divide by (1-cos x)
=∫(csc x )2-cot x csc x dx
=-cot x+csc x+c
Question 29.
Mark (√) against the correct answer in each of the following:
A. cosec x + C
B. 
C. 
D. 
Answer:
Given:
Question 30.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
=∫tan-1(tan x )dx
=∫x dx
Question 31.
Mark (√) against the correct answer in each of the following:
A.
B. 
C.
D. 
Answer:
Given:
=∫cot-1(cot x) dx
=∫x dx
Question 32.
Mark (√) against the correct answer in each of the following:
A. 
B.
C.
D.
Answer:
Given:
=∫2x dx
=x2+c
Question 33.
Mark (√) against the correct answer in each of the following:
A.
B.
C.
D.
Answer:
Given:
=∫2x dx
=x2+c
Question 34.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C.
D. 
Answer:
Given:
Question 35.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. None of these
Answer:
Given:
Question 36.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. None of these
Answer:
Given:
Question 37.
Mark (√) against the correct answer in each of the following:
A. sin x – cos x + C
B. tan x – cos x + C
C. sec x – cosec x + C
D. None of these
Answer:
Given:
=∫(tan x sec x + csc x cot x )dx
=sec x-csc x+c
Question 38.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Let x-α=t
dx=dt
=∫cos α+sin α cot t dt
=t cos α+sin α ln|sin t |+c
=(x-α) cos α+(sinα)ln|sin(x-α) |+c
=x cos α+ (sin α)ln|sin(x-α) |+c
Question 39.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. 
Answer:
Given:
Question 40.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. None of these
Answer:
Given:
Question 41.
Mark (√) against the correct answer in each of the following:
A. 
B. 
C. 
D. None of these
Answer:
Given:
