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NCERT solutions Mathematics Class 12 Part 2 chapter 7 Integrals

Chapters

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2

Chapter 7 - Integrals

Page 299

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

sin 2x

Q 1 | Page 299

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

Cos 3x

Q 2 | Page 299

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

`e^(2x)`

Q 3 | Page 299

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

(ax + b)2

Q 4 | Page 299

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

sin 2x – 4 e3x

Q 5 | Page 299

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

Q 6 | Page 299

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

Q 7 | Page 299

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

Q 8 | Page 299

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

Q 9 | Page 299

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

Q 10 | Page 299

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

Q 11 | Page 299

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

Q 12 | Page 299

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

 

Q 13 | Page 299

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

Q 14 | Page 299

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

Q 15 | Page 299

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

Q 16 | Page 299

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

Q 17 | Page 299

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

Q 18 | Page 299

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

Q 19 | Page 299

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

Q 20 | 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`

Q 21 | 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`

 

Q 22 | Page 299

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

Page 330

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

Q 1 | Page 330

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

Q 2 | Page 330

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

Q 3 | Page 330

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

Q 4 | Page 330

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

Q 5 | Page 330

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

Q 6 | Page 330

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

Q 7 | Page 330

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

Q 8 | Page 330

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

Q 9 | 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`

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`

Q 10 | 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`

Q 11 | Page 330

Page 334

Evaluate the following definite integrals as limit of sums.

`int_a^b x dx`

Q 1 | Page 334

Evaluate the following definite integrals as limit of sums.

`int_0^5 (x+1) dx`

Q 2 | Page 334

Evaluate the following definite integrals as limit of sums. 

`int_2^3 x^2 dx`

Q 3 | Page 334

Evaluate the following definite integrals as limit of sums.

`int_1^4 (x^2 - x) dx`

Q 4 | Page 334

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

Q 5 | Page 334

Evaluate the following definite integrals as limit of sums.

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

Q 6 | Page 334

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 integrals in using substitution

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

Q 3 | Page 340

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

Q 3 | Page 338

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

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_0^4 dx/(1+x^2)`

Q 10 | Page 338

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

Page 340

Evaluate the integrals in using substitution

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

Q 1 | Page 340

Evaluate the integrals in using substitution 

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

Q 2 | Page 340

Evaluate the integrals in using substitution

 `int_0^2 xsqrt(x+2)`  (Put x + 2 = `t^2`)

Q 4 | Page 340

Evaluate the integrals in using substitution

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

Q 5 | Page 340

Evaluate the integrals in using substitution

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

 

Q 6 | Page 340

Evaluate the integrals in using substitution

 `int_(-1)^1 dx/(x^2 + 2x  + 5)`

Q 7 | Page 340

Evaluate the integrals in using substitution

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

Q 8 | 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

Q 9 | Page 340

Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 1 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 2 | 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) `

Q 3 | 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)`

Q 4 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 5 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_2^8 |x - 5| dx`

Q 6 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 7 | Page 347

By using the properties of definite integrals, evaluate the integrals 

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

Q 8 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_0^2 xsqrt(2 -x)dx`

Q 9 | Page 347

By using the properties of definite integrals, evaluate the integrals 

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

Q 10 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 11 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 12 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 13 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_0^(2x) cos^5 xdx`

Q 14 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 15 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_0^pi log(1+ cos x) dx`

Q 16 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 17 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_0^4 |x - 1| dx`

Q 18 | 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`

Q 19 | 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

Q 20 | 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

Q 21 | Page 347

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

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2
S