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NCERT solutions for Class 12 Mathematics chapter 7 - Integrals

Mathematics Textbook for Class 12

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Chapters

NCERT Mathematics Class 12

Mathematics Textbook for Class 12

Chapter 7: Integrals

Chapter 7: Integrals solutions [Page 299]

Q 1 | Page 299

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

sin 2x

Q 2 | Page 299

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

Cos 3x

Q 3 | Page 299

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

`e^(2x)`

Q 4 | Page 299

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

(ax + b)2

Q 5 | Page 299

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

sin 2x – 4 e3x

Q 6 | Page 299

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

Q 7 | Page 299

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

Q 8 | Page 299

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

Q 9 | Page 299

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

Q 10 | Page 299

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

Q 11 | Page 299

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

Q 12 | Page 299

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

Q 13 | Page 299

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

 

Q 14 | Page 299

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

Q 15 | Page 299

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

Q 16 | Page 299

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

Q 17 | Page 299

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

Q 18 | Page 299

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

Q 19 | Page 299

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

Q 20 | Page 299

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

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`

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`

 

Chapter 7: Integrals solutions [Pages 304 - 305]

Q 1 | Page 304

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

Q 2 | Page 304

Integrate the functions `(log x)^2/x`

Q 3 | Page 304

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

Q 4 | Page 304

Integrate the functions sin x ⋅ sin (cos x)

Q 5 | Page 304

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

Q 6 | Page 304

Integrate the functions `sqrt(ax + b)`

Q 7 | Page 304

Integrate the functions `xsqrt(x + 2)`

Q 8 | Page 304

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

Q 9 | Page 304

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

Q 10 | Page 304

Integrate the functions `1/(x-sqrtx)`

Q 11 | 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 12 | Page 304

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

Q 13 | Page 304

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

Q 14 | Page 304

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

Q 15 | Page 304

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

Q 16 | Page 304

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

Q 17 | Page 304

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

Q 18 | Page 305

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

Q 19 | Page 305

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

 

Q 20 | Page 305

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

Q 21 | Page 305

Integrate the functions tan2(2x – 3)

Q 22 | Page 305

Integrate the functions sec2(7 – 4x)

Q 23 | Page 305

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

Q 24 | Page 305

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

Q 25 | Page 305

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

Q 26 | Page 305

Integrate the functions `cos sqrt(x)/sqrtx`

Q 27 | Page 305

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

Q 28 | Page 305

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

Q 29 | Page 305

Integrate the functions cot x log sin x

Q 30 | Page 305

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

Q 31 | Page 305

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

Q 32 | Page 305

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

Q 33 | Page 305

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

Q 34 | Page 305

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

Q 35 | Page 305

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

Q 36 | Page 305

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

Q 37 | Page 305

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

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

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

Chapter 7: Integrals solutions [Page 307]

Q 1 | Page 307

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

Q 2 | Page 307

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

Q 3 | Page 307

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

Q 4 | Page 307

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

Q 5 | Page 307

Find the integrals of the functions sin3 x cos3 x

Q 6 | Page 307

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

Q 7 | Page 307

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

Q 8 | Page 307

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

Q 9 | Page 307

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

Q 10 | Page 307

Find the integrals of the functions `sin^4 x`

Q 11 | Page 307

Find the integrals of the functions cos4 2x

Q 12 | Page 307

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

Q 13 | Page 307

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

Q 14 | Page 307

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

Q 15 | Page 307

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

Q 16 | Page 307

Find the integrals of the functions tan4x

Q 17 | Page 307

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

Q 18 | Page 307

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

Q 19 | Page 307

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

 

Q 20 | Page 307

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

Q 21 | Page 307

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

Q 22 | Page 307

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

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

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

Chapter 7: Integrals solutions [Pages 315 - 316]

Q 1 | Page 315

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

Q 2 | Page 315

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

Q 3 | Page 315

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

Q 4 | Page 315

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

Q 5 | Page 315

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

Q 6 | Page 315

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

Q 7 | Page 315

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

Q 8 | Page 315

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

Q 9 | Page 315

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

Q 10 | Page 316

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

Q 11 | Page 316

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

Q 12 | Page 316

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

Q 13 | Page 316

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

Q 14 | Page 316

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

Q 15 | Page 316

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

Q 16 | Page 316

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

Q 17 | Page 316

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

Q 18 | Page 316

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

Q 19 | Page 316

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

Q 20 | Page 316

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

Q 21 | Page 316

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

Q 22 | Page 316

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

Q 23 | Page 316

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

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

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`

Chapter 7: Integrals solutions [Pages 322 - 323]

Q 1 | Page 322

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

Q 2 | Page 322

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

Q 3 | Page 322

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

Q 4 | Page 322

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

Q 5 | Page 322

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

Q 6 | Page 322

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

Q 7 | Page 322

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

Q 8 | Page 322

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

Q 9 | Page 322

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

Q 10 | Page 322

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

Q 11 | Page 322

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

Q 12 | Page 322

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

Q 13 | Page 322

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

Q 14 | Page 322

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

Q 15 | Page 322

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

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]

Q 17 | Page 322

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

Q 18 | Page 323

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

Q 19 | Page 323

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

Q 20 | Page 323

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

Q 21 | Page 323

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

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

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`

Chapter 7: Integrals solutions [Pages 327 - 328]

Q 1 | Page 327

Integrate the functions x sin x

Q 2 | Page 327

Integrate the functions x sin 3x

Q 3 | Page 327

Integrate the functions `x^2e^x`

Q 4 | Page 327

Integrate the functions x logx

Q 5 | Page 327

Integrate the functions x log 2x

Q 6 | Page 327

Integrate the functions xlog x

Q 7 | Page 327

Integrate the functions x sin– 1x

Q 8 | Page 327

Integrate the functions x tan–1 x

Q 9 | Page 327

Integrate the functions x cos–1 x

Q 10 | Page 327

Integrate the functions (sin–1x)2

Q 11 | Page 327

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

Q 12 | Page 327

Integrate the functions x sec2 x

Q 13 | Page 327

Integrate the functions tan–1x

Q 14 | Page 327

Integrate the functions x (log x)2

Q 15 | Page 327

Integrate the functions (x2 + 1) log x

Q 16 | Page 328

Integrate the functions ex (sinx + cosx)

Q 17 | Page 328

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

Q 18 | Page 328

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

Q 19 | Page 328

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

Q 20 | Page 328

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

Q 21 | Page 328

Integrate the functions e2x sin x

Q 22 | Page 328

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

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`

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

Chapter 7: Integrals solutions [Page 330]

Q 1 | Page 330

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

Q 2 | Page 330

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

Q 3 | Page 330

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

Q 4 | Page 330

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

Q 5 | Page 330

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

Q 6 | Page 330

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

Q 7 | Page 330

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

Q 8 | Page 330

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

Q 9 | Page 330

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

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(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 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`

Chapter 7: Integrals solutions [Page 334]

Q 1 | Page 334

Evaluate the following definite integrals as limit of sums.

`int_a^b x dx`

Q 2 | Page 334

Evaluate the following definite integrals as limit of sums.

`int_0^5 (x+1) dx`

Q 3 | Page 334

Evaluate the following definite integrals as limit of sums. 

`int_2^3 x^2 dx`

Q 4 | Page 334

Evaluate the following definite integrals as limit of sums.

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

Q 5 | Page 334

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

Q 6 | Page 334

Evaluate the following definite integrals as limit of sums.

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

Chapter 7: Integrals solutions [Pages 338 - 340]

Q 1 | Page 338

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

Q 2 | Page 338

Evaluate the definite integrals `int_2^3 1/x dx`

Q 3 | Page 340

Evaluate the integrals in using substitution

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

Q 3 | Page 338

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

Q 4 | Page 338

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

Q 5 | Page 338

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

Q 6 | Page 338

Evaluate the definite integrals `int_4^5 e^x dx`

Q 7 | Page 338

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

Q 8 | Page 338

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

Q 9 | Page 338

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

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

Q 10 | Page 338

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

Q 11 | Page 338

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

Q 12 | Page 338

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

Q 13 | Page 338

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

Q 14 | Page 338

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

Q 15 | Page 338

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

Q 16 | Page 338

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

Q 17 | Page 338

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

Q 18 | Page 338

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

Q 19 | Page 338

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

Q 20 | Page 338

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

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`

Q 22 | Page 338

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

Chapter 7: Integrals solutions [Page 340]

Q 1 | Page 340

Evaluate the integrals in using substitution

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

Q 2 | Page 340

Evaluate the integrals in using substitution 

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

Q 4 | Page 340

Evaluate the integrals in using substitution

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

Q 5 | Page 340

Evaluate the integrals in using substitution

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

Q 6 | Page 340

Evaluate the integrals in using substitution

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

 

Q 7 | Page 340

Evaluate the integrals in using substitution

 `int_(-1)^1 dx/(x^2 + 2x  + 5)`

Q 8 | Page 340

Evaluate the integrals in using substitution

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

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

Chapter 7: Integrals solutions [Page 347]

Q 1 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

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` 

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) `

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)`

Q 5 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 6 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_2^8 |x - 5| dx`

Q 7 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 8 | Page 347

By using the properties of definite integrals, evaluate the integrals 

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

Q 9 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

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`

Q 11 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 12 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 13 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 14 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 15 | 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 16 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 17 | Page 347

By using the properties of definite integrals, evaluate the integrals

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

Q 18 | Page 347

By using the properties of definite integrals, evaluate the integrals

`int_0^4 |x - 1| dx`

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`

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

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

Chapter 7: Integrals solutions [Pages 352 - 354]

Q 1 | Page 352

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

Q 2 | Page 352

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

Q 3 | Page 352

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

Q 4 | Page 352

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

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]`

Q 6 | Page 352

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

Q 7 | Page 352

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

Q 8 | Page 352

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

Q 9 | Page 352

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

Q 10 | Page 352

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

Q 11 | Page 352

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

Q 12 | Page 352

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

Q 13 | Page 352

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

Q 14 | Page 352

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

Q 15 | Page 352

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

Q 16 | Page 352

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

Q 17 | Page 352

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

Q 18 | Page 352

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

Q 19 | Page 352

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

Q 20 | Page 352

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

Q 21 | Page 352

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

Q 22 | Page 352

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

Q 23 | Page 353

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

Q 24 | Page 353

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

Q 25 | Page 353

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

Q 26 | Page 353

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

Q 27 | Page 353

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

Q 28 | Page 353

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

Q 29 | Page 353

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

Q 30 | Page 353

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

Q 31 | Page 353

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

Q 32 | Page 353

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

Q 33 | Page 353

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

Q 34 | Page 353

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

Q 35 | Page 353

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

Q 36 | Page 353

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

Q 37 | Page 353

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

Q 38 | Page 353

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

Q 39 | Page 353

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

Q 40 | Page 353

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

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

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`

 

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`

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`

Chapter 7: Integrals

NCERT Mathematics Class 12

Mathematics Textbook for Class 12

NCERT solutions for Class 12 Mathematics chapter 7 - Integrals

NCERT solutions for Class 12 Maths chapter 7 (Integrals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

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

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

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