# NCERT solutions Mathematics Textbook for Class 12 Part 2 chapter 1 Integrals

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#### Where to buy

 75- 180-

### Chapter 1 - Integrals

#### Pages 304 - 305

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

Q 1 | Page 304 |

Integrate the functions (log x)^2/x

Q 2 | Page 304 |

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

Q 3 | Page 304 |

Integrate the functions sin x ⋅ sin (cos x)

Q 4 | Page 304 |

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

Q 5 | Page 304 |

Integrate the functions sqrt(ax + b)

Q 6 | Page 304 |

Integrate the functions xsqrt(x + 2)

Q 7 | Page 304 |

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

Q 8 | Page 304 |

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

Q 9 | Page 304 |

Integrate the functions 1/(x-sqrtx)

Q 10 | Page 304 |

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

Q 11 | Page 304 |

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

Q 11 | Page 304 |

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

Q 12 | Page 304 |

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

Q 13 | Page 304 |

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

Q 14 | Page 304 |

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

Q 15 | Page 304 |

Integrate the functions e^(2x+3)

Q 16 | Page 304 |

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

Q 17 | Page 304 |

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

Q 18 | Page 305 |

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

Q 19 | Page 305 |

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

Q 20 | Page 305 |

Integrate the functions tan2(2x – 3)

Q 21 | Page 305 |

Integrate the functions sec2(7 – 4x)

Q 22 | Page 305 |

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

Q 23 | Page 305 |

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

Q 24 | Page 305 |

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

Q 25 | Page 305 |

Integrate the functions cos sqrt(x)/sqrtx

Q 26 | Page 305 |

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

Q 27 | Page 305 |

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

Q 28 | Page 305 |

Integrate the functions cot x log sin x

Q 29 | Page 305 |

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

Q 30 | Page 305 |

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

Q 31 | Page 305 |

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

Q 32 | Page 305 |

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

Q 33 | Page 305 |

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

Q 34 | Page 305 |

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

Q 35 | Page 305 |

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

Q 36 | Page 305 |

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

Q 37 | 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

Q 38 | 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

Q 39 | Page 305 |

#### Page 307

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

Q 1 | Page 307 |

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

Q 2 | Page 307 |

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

Q 3 | Page 307 |

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

Q 4 | Page 307 |

Find the integrals of the functions sin3 x cos3 x

Q 5 | Page 307 |

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

Q 6 | Page 307 |

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

Q 7 | Page 307 |

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

Q 8 | Page 307 |

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

Q 9 | Page 307 |

Find the integrals of the functions sin^4 x

Q 10 | Page 307 |

Find the integrals of the functions cos4 2x

Q 11 | Page 307 |

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

Q 12 | Page 307 |

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

Q 13 | Page 307 |

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

Q 14 | Page 307 |

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

Q 15 | Page 307 |

Find the integrals of the functions tan4x

Q 16 | Page 307 |

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

Q 17 | Page 307 |

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

Q 18 | Page 307 |

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

Q 19 | Page 307 |

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

Q 20 | Page 307 |

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

Q 21 | Page 307 |

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

Q 22 | 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

Q 23 | 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

Q 24 | Page 307 |

#### Pages 315 - 316

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

Q 1 | Page 315 |

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

Q 2 | Page 315 |

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

Q 3 | Page 315 |

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

Q 4 | Page 315 |

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

Q 5 | Page 315 |

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

Q 6 | Page 315 |

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

Q 7 | Page 315 |

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

Q 8 | Page 315 |

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

Q 9 | Page 315 |

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

Q 10 | Page 316 |

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

Q 11 | Page 316 |

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

Q 12 | Page 316 |

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

Q 13 | Page 316 |

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

Q 14 | Page 316 |

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

Q 15 | Page 316 |

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

Q 16 | Page 316 |

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

Q 17 | Page 316 |

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

Q 18 | Page 316 |

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

Q 19 | Page 316 |

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

Q 20 | Page 316 |

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

Q 21 | Page 316 |

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

Q 22 | Page 316 |

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

Q 23 | 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

Q 24 | 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

Q 25 | Page 316 |

#### Pages 322 - 323

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

Q 1 | Page 322 |

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

Q 2 | Page 322 |

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

Q 3 | Page 322 |

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

Q 4 | Page 322 |

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

Q 5 | Page 322 |

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

Q 6 | Page 322 |

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

Q 7 | Page 322 |

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

Q 8 | Page 322 |

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

Q 9 | Page 322 |

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

Q 10 | Page 322 |

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

Q 11 | Page 322 |

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

Q 12 | Page 322 |

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

Q 13 | Page 322 |

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

Q 14 | Page 322 |

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

Q 15 | Page 322 |

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

Q 16 | Page 322 |

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

Q 17 | Page 322 |

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

Q 18 | Page 323 |

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

Q 19 | Page 323 |

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

Q 20 | Page 323 |

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

Q 21 | 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

Q 22 | 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

Q 23 | Page 323 |

#### Pages 327 - 328

Integrate the functions x sin x

Q 1 | Page 327 |

Integrate the functions x sin 3x

Q 2 | Page 327 |

Integrate the functions x^2e^x

Q 3 | Page 327 |

Integrate the functions x logx

Q 4 | Page 327 |

Integrate the functions x log 2x

Q 5 | Page 327 |

Integrate the functions xlog x

Q 6 | Page 327 |

Integrate the functions x sin– 1x

Q 7 | Page 327 |

Integrate the functions x tan–1 x

Q 8 | Page 327 |

Integrate the functions x cos–1 x

Q 9 | Page 327 |

Integrate the functions (sin–1x)2

Q 10 | Page 327 |

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

Q 11 | Page 327 |

Integrate the functions x sec2 x

Q 12 | Page 327 |

Integrate the functions tan–1x

Q 13 | Page 327 |

Integrate the functions x (log x)2

Q 14 | Page 327 |

Integrate the functions (x2 + 1) log x

Q 15 | Page 327 |

Integrate the functions ex (sinx + cosx)

Q 16 | Page 328 |

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

Q 17 | Page 328 |

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

Q 18 | Page 328 |

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

Q 19 | Page 328 |

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

Q 20 | Page 328 |

Integrate the functions e2x sin x

Q 21 | Page 328 |

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

Q 22 | 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

Q 23 | 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

Q 24 | Page 328 |

#### Pages 338 - 340

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

Q 1 | Page 338 |

Evaluate the definite integrals int_2^3 1/x dx

Q 2 | Page 338 |

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

Q 3 | Page 338 |

Evaluate the integrals in using substitution

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

Q 3 | Page 340 |

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

Q 4 | Page 338 |

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

Q 5 | Page 338 |

Evaluate the definite integrals int_4^5 e^x dx

Q 6 | Page 338 |

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

Q 7 | Page 338 |

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

Q 8 | Page 338 |

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

Q 9 | Page 338 |

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

Q 10 | Page 338 |

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

Q 10 | Page 340 |

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

Q 11 | Page 338 |

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

Q 12 | Page 338 |

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

Q 13 | Page 338 |

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

Q 14 | Page 338 |

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

Q 15 | Page 338 |

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

Q 16 | Page 338 |

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

Q 17 | Page 338 |

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

Q 18 | Page 338 |

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

Q 19 | Page 338 |

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

Q 20 | 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

Q 21 | Page 338 |

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

Q 22 | Page 338 |

#### Pages 352 - 354

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

Q 1 | Page 352 |

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

Q 2 | Page 352 |

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

Q 3 | Page 352 |

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

Q 4 | 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]

Q 5 | Page 352 |

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

Q 6 | Page 352 |

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

Q 7 | Page 352 |

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

Q 8 | Page 352 |

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

Q 9 | Page 352 |

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

Q 10 | Page 352 |

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

Q 11 | Page 352 |

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

Q 12 | Page 352 |

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

Q 13 | Page 352 |

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

Q 14 | Page 352 |

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

Q 15 | Page 352 |

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

Q 16 | Page 352 |

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

Q 17 | Page 352 |

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

Q 18 | Page 352 |

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

Q 19 | Page 352 |

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

Q 20 | Page 352 |

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

Q 21 | Page 352 |

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

Q 22 | Page 352 |

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

Q 23 | Page 353 |

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

Q 24 | Page 353 |

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

Q 25 | Page 353 |

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

Q 26 | Page 353 |

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

Q 27 | Page 353 |

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

Q 28 | Page 353 |

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

Q 29 | Page 353 |

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

Q 30 | Page 353 |

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

Q 31 | Page 353 |

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

Q 32 | Page 353 |

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

Q 33 | Page 353 |

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

Q 34 | Page 353 |

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

Q 35 | Page 353 |

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

Q 36 | Page 353 |

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

Q 37 | Page 353 |

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

Q 38 | Page 353 |

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

Q 39 | Page 353 |

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

Q 40 | 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

Q 41 | 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

Q 42 | Page 353 |

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

Q 43 | 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

Q 44 | Page 354 |

#### Where to buy

 75- 180-
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