# NCERT solutions for Class 12 Maths chapter 7 - Integrals [Latest edition]

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## Solutions for Chapter 7: Integrals

Below listed, you can find solutions for Chapter 7 of CBSE, Karnataka Board PUC NCERT for Class 12 Maths.

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10Exercise 7.11Exercise 7.12
Exercise 7.1 [Page 299]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1 [Page 299]

Exercise 7.1 | Q 1 | Page 299

Find an anti derivative (or integral) of the following functions by the method of inspection.

sin 2x

Exercise 7.1 | Q 2 | Page 299

Find an anti derivative (or integral) of the following functions by the method of inspection.

Cos 3x

Exercise 7.1 | Q 3 | Page 299

Find an anti derivative (or integral) of the following functions by the method of inspection.

e^(2x)

Exercise 7.1 | Q 4 | Page 299

Find an anti derivative (or integral) of the following functions by the method of inspection.

(ax + b)2

Exercise 7.1 | Q 5 | Page 299

Find an anti derivative (or integral) of the following functions by the method of inspection

sin 2x – 4 e3x

Exercise 7.1 | Q 6 | Page 299

Find the following integrals ∫(4 e3x + 1) dx

Exercise 7.1 | Q 7 | Page 299

Find the following integrals intx^2 (1 - 1/x^2)dx

Exercise 7.1 | Q 8 | Page 299

Find the following integrals ∫(ax + bx + c) dx

Exercise 7.1 | Q 9 | Page 299

Find the following integrals  int(2x^2 + e^x)dx

Exercise 7.1 | Q 10 | Page 299

Find the following integrals int(sqrtx - 1/sqrtx)^2 dx

Exercise 7.1 | Q 11 | Page 299

Find the following integrals  int (x^3 + 5x^2   -4)/x^2 dx

Exercise 7.1 | Q 12 | Page 299

Find the following integrals int (x^3 + 3x + 4)/sqrtx dx

Exercise 7.1 | Q 13 | Page 299

Find the following integrals int (x^3 - x^2 + x - 1)/(x - 1) dx

Exercise 7.1 | Q 14 | Page 299

Find the following integrals ∫(1 – x) sqrtx dx

Exercise 7.1 | Q 15 | Page 299

Find the following integrals ∫ x( 3x2 + 2x + 3) dx

Exercise 7.1 | Q 16 | Page 299

Find the following integrals int(2x- 3cos x +e^x) dx

Exercise 7.1 | Q 17 | Page 299

Find the following integrals int(2x^2 - 3sinx + 5sqrtx) dx

Exercise 7.1 | Q 18 | Page 299

Find the following integrals ∫sec x (sec x + tan x) dx

Exercise 7.1 | Q 19 | Page 299

Find the following integrals int(sec^2x)/(cosec^2x) dx

Exercise 7.1 | Q 20 | Page 299

Find the following integrals int (2 - 3 sinx)/(cos^2 x) dx

Exercise 7.1 | Q 21 | Page 299

The anti derivative of (sqrtx + 1/ sqrtx) equals

(A) 1/3 x^(1/3) + 2x^(1/2) + C

(B) 2/3x^(2/3) + 1/2 x^2 + C

(C) 2/3x^(3/2) + 2x^(1/2) +C

(D) 3/2 x^(3/2) + 1/2x^(1/2) + C

Exercise 7.1 | Q 22 | Page 299

if d/dx f(x) = 4x^3 - 3/x^4 such that f(2) = 0 Then f(x) is

(A) x^4 + 1/x^3 - 129

(B)x^3 + 1/x^4 + 129/8

(C) x^4 + 1/x^3 + 129/8

(D) x^3 + 1/x^4 - 129/8

Exercise 7.2 [Pages 304 - 305]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.2 [Pages 304 - 305]

Exercise 7.2 | Q 1 | Page 304

Integrate the functions (2x)/(1 + x^2)

Exercise 7.2 | Q 2 | Page 304

Integrate the functions (log x)^2/x

Exercise 7.2 | Q 3 | Page 304

Integrate the functions 1/(x + x log x)

Exercise 7.2 | Q 4 | Page 304

Integrate the functions sin x ⋅ sin (cos x)

Exercise 7.2 | Q 5 | Page 304

Integrate the functions sin (ax + b) cos (ax + b)

Exercise 7.2 | Q 6 | Page 304

Integrate the functions sqrt(ax + b)

Exercise 7.2 | Q 7 | Page 304

Integrate the functions xsqrt(x + 2)

Exercise 7.2 | Q 8 | Page 304

Integrate the functions xsqrt(1+ 2x^2)

Exercise 7.2 | Q 9 | Page 304

Integrate the functions (4x + 2) sqrt(x^2 + x +1)

Exercise 7.2 | Q 10 | Page 304

Integrate the functions 1/(x-sqrtx)

Exercise 7.2 | Q 11 | Page 304

Integrate the functions  x/(sqrt(x+ 4)), x > 0

Exercise 7.2 | Q 11 | Page 304

Integrate the functions  x/(sqrt(x+ 4)), x > 0

Exercise 7.2 | Q 12 | Page 304

Integrate the functions (x^3 - 1)^(1/3) x^5

Exercise 7.2 | Q 13 | Page 304

Integrate the functions x^2/(2+ 3x^3)^3

Exercise 7.2 | Q 14 | Page 304

Integrate the functions 1/(x(log x)^m),  x > 0

Exercise 7.2 | Q 15 | Page 304

Integrate the functions x/(9 - 4x^2)

Exercise 7.2 | Q 16 | Page 304

Integrate the functions e^(2x+3)

Exercise 7.2 | Q 17 | Page 304

Integrate the functions x/(e^(x^2))

Exercise 7.2 | Q 18 | Page 305

Integrate the functions e^(tan^(-1)x)/(1+x^2)

Exercise 7.2 | Q 19 | Page 305

Integrate the functions (e^(2x) - 1)/(e^(2x) + 1)

Exercise 7.2 | Q 20 | Page 305

Integrate the functions (e^(2x) -  e^(-2x))/(e^(2x) + e^(-2x))

Exercise 7.2 | Q 21 | Page 305

Integrate the functions tan2(2x – 3)

Exercise 7.2 | Q 22 | Page 305

Integrate the functions sec2(7 – 4x)

Exercise 7.2 | Q 23 | Page 305

Integrate the functions (sin^(-1) x)/(sqrt(1-x^2))

Exercise 7.2 | Q 24 | Page 305

Integrate the functions (2cosx - 3sinx)/(6cos x + 4 sin x)

Exercise 7.2 | Q 25 | Page 305

Integrate the functions 1/(cos^23 x(1- tan x)^2

Exercise 7.2 | Q 26 | Page 305

Integrate the functions cos sqrt(x)/sqrtx

Exercise 7.2 | Q 27 | Page 305

Integrate the functions sqrt(sin 2x) cos 2x

Exercise 7.2 | Q 28 | Page 305

Integrate the functions cos x /(sqrt(1+sinx))

Exercise 7.2 | Q 29 | Page 305

Integrate the functions cot x log sin x

Exercise 7.2 | Q 30 | Page 305

Integrate the functions sin x/(1+ cos x)

Exercise 7.2 | Q 31 | Page 305

Integrate the functions in (sin x)/(1+ cos x)^2

Exercise 7.2 | Q 32 | Page 305

Integrate the functions in 1/(1 + cot x)

Exercise 7.2 | Q 33 | Page 305

Integrate the functions in 1/(1 - tan x)

Exercise 7.2 | Q 34 | Page 305

Integrate the functions in sqrt(tanx)/(sinxcos x)

Exercise 7.2 | Q 35 | Page 305

Integrate the functions in (1+ log x)^2/x

Exercise 7.2 | Q 36 | Page 305

Integrate the functions in ((x+1)(x + logx)^2)/x

Exercise 7.2 | Q 37 | Page 305

Integrate the functions in (x^3 sin(tan^(-1) x^4))/(1 + x^8)

Exercise 7.2 | Q 38 | Page 305

Choose the correct answer int (10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx equals

(A) 10x – x10 + C

(B) 10x + x10 + C

(C) (10x – x10)–1 + C

(D) log (10x + x10) + C

Exercise 7.2 | Q 39 | Page 305

Choose the correct answer int (dx)/(sin^2 x cos^2 x) equals

(A) tan x + cot x + C

(B) tan x – cot x + C

(C) tan x cot x + C

(D) tan x – cot 2x + C

Exercise 7.3 [Page 307]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.3 [Page 307]

Exercise 7.3 | Q 1 | Page 307

Find the integrals of the functions sin2 (2x + 5)

Exercise 7.3 | Q 2 | Page 307

Find the integrals of the functions sin 3x cos 4x

Exercise 7.3 | Q 3 | Page 307

Find the integrals of the functions cos 2x cos 4x cos 6x

Exercise 7.3 | Q 4 | Page 307

Find the integrals of the functions sin3 (2x + 1)

Exercise 7.3 | Q 5 | Page 307

Find the integrals of the functions sin3 x cos3 x

Exercise 7.3 | Q 6 | Page 307

Find the integrals of the functions sin x sin 2x sin 3x

Exercise 7.3 | Q 7 | Page 307

Find the integrals of the functions sin 4x sin 8x

Exercise 7.3 | Q 8 | Page 307

Find the integrals of the functions (1- cosx)/(1 +  cos x)

Exercise 7.3 | Q 9 | Page 307

Find the integrals of the functions  cos x/(1 + cos x)

Exercise 7.3 | Q 10 | Page 307

Find the integrals of the functions sin^4 x

Exercise 7.3 | Q 11 | Page 307

Find the integrals of the functions cos4 2x

Exercise 7.3 | Q 12 | Page 307

Find the integrals of the functions (sin^2 x)/(1 + cos x)

Exercise 7.3 | Q 13 | Page 307

Find the integrals of the functions  (cos 2x - cos 2 alpha)/(cos x - cos alpha)

Exercise 7.3 | Q 14 | Page 307

Find the integrals of the functions  (cos x -  sinx)/(1+sin 2x)

Exercise 7.3 | Q 15 | Page 307

Find the integrals of the functions tan3 2x sec 2x

Exercise 7.3 | Q 16 | Page 307

Find the integrals of the functions tan4x

Exercise 7.3 | Q 17 | Page 307

Find the integrals of the functions (sin^3 x + cos^3 x)/(sin^2x cos^2 x)

Exercise 7.3 | Q 18 | Page 307

Find the integrals of the functions (cos 2x+ 2sin^2x)/(cos^2 x)

Exercise 7.3 | Q 19 | Page 307

Find the integrals of the functions 1/(sin xcos^3 x)

Exercise 7.3 | Q 20 | Page 307

Find the integrals of the functions (cos 2x)/(cos x + sin x)^2

Exercise 7.3 | Q 21 | Page 307

Find the integrals of the functions sin−1 (cos x)

Exercise 7.3 | Q 22 | Page 307

Find the integrals of the functions  1/(cos(x - a) cos(x - b))

Exercise 7.3 | Q 23 | Page 307

Choose the correct answer int (sin^2x - cos^2 x)/(sin^2 x cos^2 x) dx

A. tan x + cot x + C

B. tan x + cosec x + C

C. − tan x + cot x + C

D. tan x + sec x + C

Exercise 7.3 | Q 24 | Page 307

Choose the correct answer int (e^x(1 +x))/cos^2(e^x x) dx

A. − cot (exx) + C

B. tan (xex) + C

C. tan (ex) + C

D. cot (ex) + C

Exercise 7.4 [Pages 315 - 316]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.4 [Pages 315 - 316]

Exercise 7.4 | Q 1 | Page 315

Integrate the functions (3x^2)/(x^6 + 1)

Exercise 7.4 | Q 2 | Page 315

Integrate the functions 1/sqrt(1+4x^2)

Exercise 7.4 | Q 3 | Page 315

Integrate the functions 1/sqrt((2-x)^2 + 1)

Exercise 7.4 | Q 4 | Page 315

Integrate the functions 1/sqrt(9 - 25x^2)

Exercise 7.4 | Q 5 | Page 315

Integrate the functions (3x)/(1+ 2x^4)

Exercise 7.4 | Q 6 | Page 315

Integrate the functions x^2/(1 - x^6)

Exercise 7.4 | Q 7 | Page 315

Integrate the functions (x - 1)/sqrt(x^2 - 1)

Exercise 7.4 | Q 8 | Page 315

Integrate the functions x^2/sqrt(x^6 + a^6)

Exercise 7.4 | Q 9 | Page 315

Integrate the functions (sec^2 x)/sqrt(tan^2 x + 4)

Exercise 7.4 | Q 10 | Page 316

Integrate the functions 1/sqrt(x^2 +2x + 2)

Exercise 7.4 | Q 11 | Page 316

Integrate the functions 1/sqrt(9x^2 + 6x + 5)

Exercise 7.4 | Q 12 | Page 316

Integrate the functions  1/sqrt(7 - 6x - x^2)

Exercise 7.4 | Q 13 | Page 316

Integrate the functions 1/sqrt((x -1)(x - 2))

Exercise 7.4 | Q 14 | Page 316

Integrate the functions 1/sqrt(8+3x  - x^2)

Exercise 7.4 | Q 15 | Page 316

Integrate the functions 1/sqrt((x - a)(x - b))

Exercise 7.4 | Q 16 | Page 316

Integrate the functions (4x+ 1)/sqrt(2x^2 + x - 3)

Exercise 7.4 | Q 17 | Page 316

Integrate the functions (x + 2)/sqrt(x^2 -1)

Exercise 7.4 | Q 18 | Page 316

Integrate the functions (5x - 2)/(1 + 2x + 3x^2)

Exercise 7.4 | Q 19 | Page 316

Integrate the functions (6x + 7)/sqrt((x - 5)(x - 4))

Exercise 7.4 | Q 20 | Page 316

Integrate the functions (x + 2)/sqrt(4x - x^2)

Exercise 7.4 | Q 21 | Page 316

Integrate the functions (x+2)/sqrt(x^2 + 2x + 3)

Exercise 7.4 | Q 22 | Page 316

Integrate the functions (x + 3)/(x^2 - 2x - 5)

Exercise 7.4 | Q 23 | Page 316

Integrate the functions (5x + 3)/sqrt(x^2 + 4x + 10)

Exercise 7.4 | Q 24 | Page 316

Choose the correct answer int dx/(x^2 + 2x + 2) equals

A. x tan−1 (x + 1) + C

B. tan− 1 (x + 1) + C

C. (x + 1) tan−1 x + C

D. tan−1 x + C

Exercise 7.4 | Q 25 | Page 316

Choose the correct answer int dx/sqrt(9x - 4x^2) equals

(A) 1/9 sin^(-1) ((9x -8)/8) + C

(B) 1/2 sin^(-1) ((8x -9)/9) + C

(C) 1/3 sin^(-1) ((9x - 8)/8) + C

(D) 1/2 sin^(-1) ((9x - 8)/9) + C

Exercise 7.5 [Pages 322 - 323]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.5 [Pages 322 - 323]

Exercise 7.5 | Q 1 | Page 322

Integrate the rational functions x/((x +  1)(x+ 2))

Exercise 7.5 | Q 2 | Page 322

Integrate the rational functions 1/(x^2 - 9)

Exercise 7.5 | Q 3 | Page 322

Integrate the rational functions (3x - 1)/((x - 1)(x - 2)(x - 3))

Exercise 7.5 | Q 4 | Page 322

Integrate the rational functions x/((x-1)(x- 2)(x - 3))

Exercise 7.5 | Q 5 | Page 322

Integrate the rational functions (2x)/(x^2 + 3x + 2)

Exercise 7.5 | Q 6 | Page 322

Integrate the rational functions (1 - x^2)/(x(1-2x))

Exercise 7.5 | Q 7 | Page 322

Integrate the rational functions x/((x^2+1)(x - 1))

Exercise 7.5 | Q 8 | Page 322

Integrate the rational functions x/((x -1)^2 (x+ 2))

Exercise 7.5 | Q 9 | Page 322

Integrate the rational functions (3x + 5)/(x^3 - x^2 - x + 1)

Exercise 7.5 | Q 10 | Page 322

Integrate the rational functions (2x - 3)/((x^2 -1)(2x + 3))

Exercise 7.5 | Q 11 | Page 322

Integrate the rational functions (5x)/((x + 1)(x^2 - 4))

Exercise 7.5 | Q 12 | Page 322

Integrate the rational functions (x^2 + x + 1)/(x^2 -1)

Exercise 7.5 | Q 13 | Page 322

Integrate the rational functions 2/((1-x)(1+x^2))

Exercise 7.5 | Q 14 | Page 322

Integrate the rational functions (3x -1)/(x + 2)^2

Exercise 7.5 | Q 15 | Page 322

Integrate the rational functions 1/(x^4 - 1)

Exercise 7.5 | Q 16 | Page 322

Integrate the rational functions 1/(x(x^n + 1)) [Hint: multiply numerator and denominator by xn − 1 and put xn = t]

Exercise 7.5 | Q 17 | Page 322

Integrate the rational functions  (cos x)/((1-sinx)(2 - sin x)) [Hint: Put sin x = t]

Exercise 7.5 | Q 18 | Page 323

Integrate the rational functions ((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))

Exercise 7.5 | Q 19 | Page 323

Integrate the rational functions  (2x)/((x^2 + 1)(x^2 + 3))

Exercise 7.5 | Q 20 | Page 323

Integrate the rational functions 1/(x(x^4 - 1))

Exercise 7.5 | Q 21 | Page 323

Integrate the rational functions 1/(e^x -1)[Hint: Put ex = t]

Exercise 7.5 | Q 22 | Page 323

Choose the correct answer int (xdx)/((x - 1)(x - 2)) equals

A. log |(x - 1)^2/(x -2)| + C

B. log |(x-2)^2/(x -1)| +C

C. log|((x- 1)/(x- 2))^2| + C

D. log|(x - 1)(x - 2)| + C

Exercise 7.5 | Q 23 | Page 323

Choose the correct answer int (dx)/(x(x^2 + 1)) equal

A. log |x| - 1/2 log (x^2 + 1) + C

B. log |x| + 1/2 log(x^2 + 1) + C

C. - log|x| + 1/2 log (x^2 + 1) + C

D. 1/2 log|x| + log(x^2 + 1)+ C

Exercise 7.6 [Pages 327 - 328]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.6 [Pages 327 - 328]

Exercise 7.6 | Q 1 | Page 327

Integrate the functions x sin x

Exercise 7.6 | Q 2 | Page 327

Integrate the functions x sin 3x

Exercise 7.6 | Q 3 | Page 327

Integrate the functions x^2e^x

Exercise 7.6 | Q 4 | Page 327

Integrate the functions x logx

Exercise 7.6 | Q 5 | Page 327

Integrate the functions x log 2x

Exercise 7.6 | Q 6 | Page 327

Integrate the functions xlog x

Exercise 7.6 | Q 7 | Page 327

Integrate the functions x sin– 1x

Exercise 7.6 | Q 8 | Page 327

Integrate the functions x tan–1 x

Exercise 7.6 | Q 9 | Page 327

Integrate the functions x cos–1 x

Exercise 7.6 | Q 10 | Page 327

Integrate the functions (sin–1x)2

Exercise 7.6 | Q 11 | Page 327

Integrate the functions (x cos^(-1) x)/sqrt(1-x^2)

Exercise 7.6 | Q 12 | Page 327

Integrate the functions x sec2 x

Exercise 7.6 | Q 13 | Page 327

Integrate the functions tan–1x

Exercise 7.6 | Q 14 | Page 327

Integrate the functions x (log x)2

Exercise 7.6 | Q 15 | Page 327

Integrate the functions (x2 + 1) log x

Exercise 7.6 | Q 16 | Page 328

Integrate the functions ex (sinx + cosx)

Exercise 7.6 | Q 17 | Page 328

Integrate the functions (xe^x)/(1+x)^2

Exercise 7.6 | Q 18 | Page 328

Integrate the functions e^x (1 + sin x)/(1+cos x)

Exercise 7.6 | Q 19 | Page 328

Integrate the functions e^x (1/x - 1/x^2)

Exercise 7.6 | Q 20 | Page 328

Integrate the functions ((x- 3)e^x)/(x - 1)^3

Exercise 7.6 | Q 21 | Page 328

Integrate the functions e2x sin x

Exercise 7.6 | Q 22 | Page 328

Integrate the functions sin^(-1) ((2x)/(1+x^2))

Exercise 7.6 | Q 23 | Page 328

Choose the correct answer intx^2 e^(x^3) dx equals

(A) 1/3 e^(x^3) + C

(B) 1/3 e^(x^2) + C

(C) 1/2 e^(x^3) +C

(D) 1/3 e^(x^2) + C

Exercise 7.6 | Q 24 | Page 328

Choose the correct answer int e^x sec x (1 +   tan x) dx equals

(A) ex cos x + C

(B) ex sec x + C

(C) ex sin x + C

(D) ex tan x + C

Exercise 7.7 [Page 330]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.7 [Page 330]

Exercise 7.7 | Q 1 | Page 330

Integrate the functions sqrt(4 - x^2)

Exercise 7.7 | Q 2 | Page 330

Integrate the functions sqrt(1- 4x^2)

Exercise 7.7 | Q 3 | Page 330

Integrate the functions sqrt(x^2 + 4x + 6)

Exercise 7.7 | Q 4 | Page 330

Integrate the functions sqrt(x^2 + 4x +1)

Exercise 7.7 | Q 5 | Page 330

Integrate the functions sqrt(1-4x - x^2)

Exercise 7.7 | Q 6 | Page 330

Integrate the functions sqrt(x^2 + 4x - 5)

Exercise 7.7 | Q 7 | Page 330

Integrate the functions sqrt(1+ 3x - x^2)

Exercise 7.7 | Q 8 | Page 330

Integrate the functions sqrt(x^2 + 3x)

Exercise 7.7 | Q 9 | Page 330

Integrate the functions sqrt(1+ x^2/9)

Exercise 7.7 | Q 10 | Page 330

Choose the correct answer int sqrt(1+ x^2) dx is equal

(A) x/2 sqrt(1+ x^2) +  1/2log |(x + sqrt(1+ x^2)) + C|

(B) 2/3 (1+ x^2)^(3/2) + C

(C) 2/3x(1 + x^2)^(3/2) + C

(D) x^2/2 sqrt(1+x^2) + 1/2 x^2 log|x + sqrt(1+ x^2)|+ C

Exercise 7.7 | Q 10 | Page 330

Choose the correct answer int sqrt(1+ x^2) dx is equal

(A) x/2 sqrt(1+ x^2) +  1/2log |(x + sqrt(1+ x^2)) + C|

(B) 2/3 (1+ x^2)^(3/2) + C

(C) 2/3x(1 + x^2)^(3/2) + C

(D) x^2/2 sqrt(1+x^2) + 1/2 x^2 log|x + sqrt(1+ x^2)|+ C

Exercise 7.7 | Q 11 | Page 330

Choose the correct answer int sqrt(x^2 - 8x + 7) dx is equal to

(A) 1/2 (x - 4) sqrt(x^2 - 8x + 7) + 9log |x - 4 +  sqrt(x^2 - 8x + 7)| + C

(B) 1/2( x- 4) sqrt(x^2-8x + 7) + 9log|x + 4 + sqrt(x^2 - 8x + 7)| + C

(C) 1/2 (x - 4) sqrt(x^2 - 8x + 7) - 3sqrt2 log |x - 4 + sqrt(x^2 - 8x + 7)| + C

(D) 1/2 (x - 4) sqrt(x^2 - 8x + 7) - 9/2 log |x - 4 +  sqrt(x^2 - 8x + 7)|  + C

Exercise 7.8 [Page 334]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.8 [Page 334]

Exercise 7.8 | Q 1 | Page 334

Evaluate the following definite integrals as limit of sums.

int_a^b x dx

Exercise 7.8 | Q 2 | Page 334

Evaluate the following definite integrals as limit of sums.

int_0^5 (x+1) dx

Exercise 7.8 | Q 3 | Page 334

Evaluate the following definite integrals as limit of sums.

int_2^3 x^2 dx

Exercise 7.8 | Q 4 | Page 334

Evaluate the following definite integrals as limit of sums.

int_1^4 (x^2 - x) dx

Exercise 7.8 | Q 5 | Page 334

Evaluate the following definite integrals as limit of sums int_(-1)^1 e^x dx

Exercise 7.8 | Q 6 | Page 334

Evaluate the following definite integrals as limit of sums.

int_0^4 (x + e^(2x)) dx

Exercise 7.9 [Page 338]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.9 [Page 338]

Exercise 7.9 | Q 1 | Page 338

Evaluate the definite integrals int_(-1)^1 (x + 1)dx

Exercise 7.9 | Q 2 | Page 338

Evaluate the definite integrals int_2^3 1/x dx

Exercise 7.9 | Q 3 | Page 340

Evaluate the integrals in using substitution

int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx

Exercise 7.9 | Q 3 | Page 338

Evaluate the definite integrals  int_1^2 (4x^3 - 5x^2 + 6x + 9)

Exercise 7.9 | Q 4 | Page 338

Evaluate the definite integrals int_0^(x/4) sin2xdx

Exercise 7.9 | Q 5 | Page 338

Evaluate the definite integrals int_0^(x/2) xos 2x dx

Exercise 7.9 | Q 6 | Page 338

Evaluate the definite integrals int_4^5 e^x dx

Exercise 7.9 | Q 7 | Page 338

Evaluate the definite integrals int_0^(x/4) tan x dx

Exercise 7.9 | Q 8 | Page 338

Evaluate the definite integrals int_(pi/6)^(pi/4) cosec x dx

Exercise 7.9 | Q 9 | Page 338

Evaluate the definite integrals int_0^4 dx/sqrt(1-x^2)

Exercise 7.9 | Q 10 | Page 338

Evaluate the definite integrals int_0^4 dx/(1+x^2)

Exercise 7.9 | Q 10 | Page 340

if f(x) = int_0^pi t sin t dt, then f' (x) is

A. cos x + x sin x

B. x sin x

C. x cos x

D. sin x cos x

Exercise 7.9 | Q 11 | Page 338

Evaluate the definite integrals int_2^3 dx/(x^2 - 1)

Exercise 7.9 | Q 12 | Page 338

Evaluate the definite integrals int_0^(pi/2) cos^2 xdx

Exercise 7.9 | Q 13 | Page 338

Evaluate the definite integrals int_2^3 (xdx)/(x^2 + 1)

Exercise 7.9 | Q 14 | Page 338

Evaluate the definite integrals int_0^1 (2x + 3)/(5x^2 + 1) dx

Exercise 7.9 | Q 15 | Page 338

Evaluate the definite integrals int_0^1 x e^(x^2) dx

Exercise 7.9 | Q 16 | Page 338

Evaluate the definite integrals int_1^2 (5x^2)/(x^2 + 4x + 3)

Exercise 7.9 | Q 17 | Page 338

Evaluate the definite integrals int_0^(pi/4) (2 sec^2 x + x^3  + 2) dx

Exercise 7.9 | Q 18 | Page 338

Evaluate the definite integrals int_0^Pi (sin^2 x/2 - cos^2 x/2) dx

Exercise 7.9 | Q 19 | Page 338

Evaluate the definite integrals int_0^2 (6x +3)/(x^2 + 4) dx

Exercise 7.9 | Q 20 | Page 338

Evaluate the definite integrals int_0^1 (xe^x + sin  (pix)/4)

Exercise 7.9 | Q 21 | Page 338

Choose the correct answer int_1^(sqrt3)dx/(1+x^2)  equals

A. pi/3

B. (2pi)/3

C. pi/6

D. pi/12

Exercise 7.9 | Q 22 | Page 338

Choose the correct answer int_0^(2/3) dx/(4+9x^2) equals is

Exercise 7.10 [Page 340]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.10 [Page 340]

Exercise 7.10 | Q 1 | Page 340

Evaluate the integrals in using substitution

int_0^1 x/(x^2 +1)dx

Exercise 7.10 | Q 2 | Page 340

Evaluate the integrals in using substitution

int_0^(pi/2) sqrt(sin phi) cos^5 phidphi

Exercise 7.10 | Q 4 | Page 340

Evaluate the integrals in using substitution

int_0^2 xsqrt(x+2)  (Put x + 2 = t^2)

Exercise 7.10 | Q 5 | Page 340

Evaluate the integrals in using substitution

int_0^(pi/2) (sin x)/(1+ cos^2 x) dx

Exercise 7.10 | Q 6 | Page 340

Evaluate the integrals in using substitution

int_0^2 dx/(x + 4 - x^2)

Exercise 7.10 | Q 7 | Page 340

Evaluate the integrals in using substitution

int_(-1)^1 dx/(x^2 + 2x  + 5)

Exercise 7.10 | Q 8 | Page 340

Evaluate the integrals in using substitution

int_1^2 (1/x- 1/(2x^2))e^(2x) dx

Exercise 7.10 | Q 9 | Page 340

The value of the integral int_(1/3)^4 ((x- x^3)^(1/3))/x^4 dx is

A. 6

B. 0

C. 3

D. 4

Exercise 7.11 [Page 347]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.11 [Page 347]

Exercise 7.11 | Q 1 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2) cos^2 x dx

Exercise 7.11 | Q 2 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx

Exercise 7.11 | Q 3 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2) (sin^(3/2) xdx)/(sin^(3/2) x + cos^(3/2)x)

Exercise 7.11 | Q 4 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2)  (cos^5  xdx)/(sin^5 x + cos^5 x)

Exercise 7.11 | Q 5 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_(-5)^5 | x + 2| dx

Exercise 7.11 | Q 6 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_2^8 |x - 5| dx

Exercise 7.11 | Q 7 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^4 x(1-x)^n dx

Exercise 7.11 | Q 8 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/4) log (1+ tan x) dx

Exercise 7.11 | Q 9 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^2 xsqrt(2 -x)dx

Exercise 7.11 | Q 10 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2) (2log sin x - log sin 2x)dx

Exercise 7.11 | Q 11 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_((-pi)/2)^(pi/2) sin^2 x dx

Exercise 7.11 | Q 12 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^pi (xdx)/(1+ sin x)

Exercise 7.11 | Q 13 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_(pi/2)^(pi/2) sin^7 x dx

Exercise 7.11 | Q 14 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(2x) cos^5 xdx

Exercise 7.11 | Q 15 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^(pi/2) (sin "x" - cos "x")/(1+sin"x" cos "x") "dx"

Exercise 7.11 | Q 16 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^pi log(1+ cos x) dx

Exercise 7.11 | Q 17 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^a  sqrtx/(sqrtx + sqrt(a-x))   dx

Exercise 7.11 | Q 18 | Page 347

By using the properties of definite integrals, evaluate the integrals

int_0^4 |x - 1| dx

Exercise 7.11 | Q 19 | Page 347

Show that int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4

Exercise 7.11 | Q 20 | Page 347

Choose the correct The value of int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx  is

A. 0

B. 2

C. π

D. 1

Exercise 7.11 | Q 21 | Page 347

Choose the correct The value of int_0^(pi/2) log  ((4+ 3sinx)/(4+3cos x)) dx is

A. 2

B. 3/4

C. 0

D. -2

Exercise 7.12 [Pages 352 - 354]

### NCERT solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.12 [Pages 352 - 354]

Exercise 7.12 | Q 1 | Page 352

Integrate the functions 1/(x - x^3)

Exercise 7.12 | Q 2 | Page 352

Integrate the functions 1/(sqrt(x+a) + sqrt(x+b))

Exercise 7.12 | Q 3 | Page 352

Integrate the functions 1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]

Exercise 7.12 | Q 4 | Page 352

Integrate the functions 1/(x^2(x^4 + 1)^(3/4))

Exercise 7.12 | Q 5 | Page 352

Integrate the functions

1/(x^(1/2) + x^(1/3))  ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))) put x = t^6]

Exercise 7.12 | Q 6 | Page 352

Integrate the functions  (5x)/((x+1)(x^2 +9))

Exercise 7.12 | Q 7 | Page 352

Integrate the functions sinx/(sin (x - a))

Exercise 7.12 | Q 8 | Page 352

Integrate the functions (e^(5log x) -  e^(4log x))/(e^(3log x) - e^(2log x))

Exercise 7.12 | Q 9 | Page 352

Integrate the functions cos x/sqrt(4 - sin^2 x)

Exercise 7.12 | Q 10 | Page 352

Integrate the functions (sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)

Exercise 7.12 | Q 11 | Page 352

Integrate the functions 1/(cos (x+a) cos(x+b))

Exercise 7.12 | Q 12 | Page 352

Integrate the functions x^3/(sqrt(1-x^8)

Exercise 7.12 | Q 13 | Page 352

Integrate the functions  e^x/((1+e^x)(2+e^x))

Exercise 7.12 | Q 14 | Page 352

Integrate the functions  1/((x^2 + 1)(x^2 + 4))

Exercise 7.12 | Q 15 | Page 352

Integrate the functions cos^3 xe^(log sinx)

Exercise 7.12 | Q 16 | Page 352

Integrate the functions e^(3log x) (x^4 + 1)^(-1)

Exercise 7.12 | Q 17 | Page 352

Integrate the functions f'(ax +b)[f(ax +b)]^n

Exercise 7.12 | Q 18 | Page 352

Integrate the functions 1/sqrt(sin^3 x sin(x + a))

Exercise 7.12 | Q 19 | Page 352

Integrate the functions (sin^(-1) sqrtx - cos^(-1) sqrtx)/ (sin^(-1) sqrtx + cos^(-1) sqrtx) , x in [0,1]

Exercise 7.12 | Q 20 | Page 352

Integrate the functions sqrt((1-sqrtx)/(1+sqrtx))

Exercise 7.12 | Q 21 | Page 352

Integrate the functions (2+ sin 2x)/(1+ cos 2x) e^x

Exercise 7.12 | Q 22 | Page 352

Integrate the functions (x^2 + x + 1)/((x + 1)^2 (x + 2))

Exercise 7.12 | Q 23 | Page 353

Integrate the functions tan^(-1) sqrt((1-x)/(1+x))

Exercise 7.12 | Q 24 | Page 353

Integrate the functions (sqrt(x^2 +1) [log(x^2 + 1) - 2log x])/x^4

Exercise 7.12 | Q 25 | Page 353

Evaluate the definite integrals int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx

Exercise 7.12 | Q 26 | Page 353

Evaluate the definite integrals int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)dx

Exercise 7.12 | Q 27 | Page 353

Evaluate the definite integrals  int_0^(pi/2) (cos^2 x dx)/(xos^2 x + 4 sin^2 x)

Exercise 7.12 | Q 28 | Page 353

Evaluate the definite integrals int_(pi/6)^(pi/3)  (sin x +   cosx)/sqrt(sin 2x) dx

Exercise 7.12 | Q 29 | Page 353

Evaluate the definite integrals int_0^1 dx/(sqrt(1+x) - sqrtx)

Exercise 7.12 | Q 30 | Page 353

Evaluate the definite integrals int_0^(pi/4) (sin x +  cos x)/(9+16sin 2x) dx

Exercise 7.12 | Q 31 | Page 353

Evaluate the definite integrals int_0^(pi/2) sin 2x tan^(-1) (sinx) dx

Exercise 7.12 | Q 32 | Page 353

Evaluate the definite integrals int_0^pi (x tan x)/(sec x + tan x) dx

Exercise 7.12 | Q 33 | Page 353

Evaluate the definite integrals int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx

Exercise 7.12 | Q 34 | Page 353

Prove the following int_1^3 dx/(x^2(x +1)) = 2/3 + log 2/3

Exercise 7.12 | Q 35 | Page 353

Prove the following int_0^4 xe^x dx = 1

Exercise 7.12 | Q 36 | Page 353

Prove the following int_(-1)^1 x^17 cos^4 xdx = 0

Exercise 7.12 | Q 37 | Page 353

Prove the following int_0^(pi/2) sin^3 xdx = 2/3

Exercise 7.12 | Q 38 | Page 353

Prove the following int_0^(pi/4) 2 tan^3 xdx = 1 - log 2

Exercise 7.12 | Q 39 | Page 353

Prove the following int_0^1sin^(-1) xdx = pi/2 - 1

Exercise 7.12 | Q 40 | Page 353

Evaluate  int_0^1 e^(2-3x) dx as a limit of a sum.

Exercise 7.12 | Q 41 | Page 353

Choose the correct int dx/(e^x + e^(-x)) is equal to

(A) tan–1 (ex) + C

(B) tan–1 (e–x) + C

(C) log (ex – e–x) + C

(D) log (ex + e–x) + C

Exercise 7.12 | Q 42 | Page 353

Choose the correct answers int (cos 2x)/(sin x + cos x)^2dx is equal to

A (-1)/(sin x + cos x) + C

(B) log |sin x + cos x | + C

(B) log |sin x + cos x | + C

(D) 1/(sin x - cos x)^2

Exercise 7.12 | Q 43 | Page 354

Choose the correct answers If f (a + b – x) = f (x), then int_a^b x f(x )dx is equal

A (a+b)/2 int_a^b f(b-x) dx

B  (a+b)/2 int_a^b f(b+x) dx

C (b-a)/2 int_a^b f(x) dx

D (a+b)/2 int_a^b f(x) dx

Exercise 7.12 | Q 44 | Page 354

Choose the correct answers The value of int_0^1 tan^(-1)  (2x -1)/(1+x - x^2) dx is

(A) 1

(B) 0

(C) –1

(D) pi/4

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## Solutions for Chapter 7: Integrals

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10Exercise 7.11Exercise 7.12

## NCERT solutions for Class 12 Maths chapter 7 - Integrals

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 7 (Integrals) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Class 12 Maths chapter 7 Integrals are Definite Integrals Problems, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Integrals of Some Particular Functions, Indefinite Integral by Inspection, Some Properties of Indefinite Integral, Integration Using Trigonometric Identities, Introduction of Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Fundamental Theorem of Calculus, Definite Integral as the Limit of a Sum, Evaluation of Simple Integrals of the Following Types and Problems, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions, Methods of Integration: Integration by Substitution, Integration as an Inverse Process of Differentiation, Geometrical Interpretation of Indefinite Integrals.

Using NCERT Class 12 Maths solutions Integrals exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Solutions are essential questions that can be asked in the final exam. Maximum CBSE, Karnataka Board PUC Class 12 Maths students prefer NCERT Textbook Solutions to score more in exams.

Get the free view of Chapter 7, Integrals Class 12 Maths additional questions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC, and you can use Shaalaa.com to keep it handy for your exam preparation.

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