Class 12 Maths NCERT Solutions

# NCERT Class 12 Maths Solutions Chapter 7 – Integrals

NCERT Class 12 Maths Solutions Chapter 7 – Integrals: Hello dear students. In this page, you will get the full solution of Integrals. So solve the questions of this chapter at your home.

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## NCERT Class 12 Maths Solutions Chapter 7 – Integrals

The topics and sub-topics included in the Integrals chapter are the following:

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals

#### Question 1:

sin 2x

The anti derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that,

Therefore, the anti derivative of

#### Question 2:

Cos 3x

The anti derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that,

Therefore, the anti derivative of .

#### Question 3:

e2x

The anti derivative of e2is the function of x whose derivative is e2x.

It is known that,

Therefore, the anti derivative of .

#### Question 4:

The anti derivative of

is the function of whose derivative is .

It is known that,

Therefore, the anti derivative of .

#### Question 5:

The anti derivative of  is the function of x whose derivative is .

It is known that,

Therefore, the anti derivative of  is .

#### Question 13:

On dividing, we obtain

#### Question 21:

The anti derivative of equals

(A)  (B)

(C)  (D)

Hence, the correct answer is C.

#### Question 22:

If such that f(2) = 0, then f(x) is

(A)  (B)

(C)  (D)

It is given that,

∴Anti derivative of

Also,

Hence, the correct answer is A.

Let t

∴2x dx = dt

Let log |x| = t

∴

Let 1 + log t

∴

#### Question 4:

sin x ⋅ sin (cos x)

sin x ⋅ sin (cos x)

Let cos x = t

∴ −sin x dx = dt

Let

Let ax + b = t

Let

∴ dx = dt

Let 1 + 2x2 = t

∴ 4xdx = dt

#### Question 9:

Let

∴ (2x + 1)dx = dt

Let

#### Question 11:

Let I=∫xx+4 dxput x+4=t⇒dx=dtNow, I=∫t-4tdt=∫t-4t-1/2dt=23t3/2-42t1/2+C=23.t.t1/2-8t1/2+C=23x+4x+4-8x+4+C=23xx+4+83x+4-8x+4+C=23xx+4-163x+4+C=23x+4x-8+C

Let

∴

Let

∴ 9x2 dx = dt

Let log x = t

∴

Let

∴ −8x dx = dt

Let

∴ 2dx = dt

Let

∴ 2xdx = dt

Let

∴

#### Question 19:

Dividing numerator and denominator by ex, we obtain

Let

∴

Let

∴

#### Question 21:

Let 2x − 3 = t

∴ 2dx = dt

⇒∫tan22x-3dx = ∫sec22x-3 – 1dx=∫sec2t- 1dt2= 12∫sec2t dt – ∫1dt= 12tant – t + C= 12tan2x-3 – 2x-3 + C

Let 7 − 4x = t

∴ −4dx = dt

Let

∴

Let

∴

Let

∴

Let

∴

Let sin 2x = t

∴

Let

∴ cos x dx = dt

#### Question 29:

cot x log sin x

Let log sin x = t

#### Question 30:

Let 1 + cos x = t

∴ −sin x dx = dt

#### Question 31:

Let 1 + cos x = t

∴ −sin x dx = dt

#### Question 32:

Let sin x + cos x = t ⇒ (cos x − sin xdx = dt

#### Question 33:

Put cos x − sin x = t ⇒ (−sin x − cos xdx = dt

#### Question 35:

Let 1 + log x = t

∴

Let

∴

#### Question 37:

Let x4 = t

∴ 4x3 dx = dt

Let

From (1), we obtain

#### Question 38:

equals

Let

∴

Hence, the correct answer is D.

#### Question 39:

equals

A.

B.

C.

D.

Hence, the correct answer is B.

#### Question 2:

It is known that,

#### Question 3:

cos 2x cos 4x cos 6x

It is known that,

sin3 (2x + 1)

Let

sin3 x cos3 x

#### Question 6:

sin x sin 2x sin 3x

It is known that,

#### Question 7:

sin 4x sin 8x

It is known that,

sin A . sin B = 12cosA-B-cosA+B∴∫sin4x sin8x dx=∫12cos4x-8x-cos4x+8xdx=12∫cos-4x-cos12xdx=12∫cos4x-cos12xdx=12sin4x4-sin12x12+C

sin4 x

cos4 2x

#### Question 16:

tan4x

From equation (1), we obtain

#### Question 19:

1sinxcos3x=sin2x+cos2xsinxcos3x=sinxcos3x+1sinxcosx

⇒1sinxcos3x=tanxsec2x+1cos2xsinxcosxcos2x=tanxsec2x+sec2xtanx

#### Question 21:

sin−1 (cos x)

It is known that,

Substituting in equation (1), we obtain

#### Question 23:

is equal to

A. tan x + cot x + C

B. tan x + cosec x + C

C. − tan x + cot x + C

D. tan x + sec x + C

Hence, the correct answer is A.

#### Question 24:

equals

A. âˆ’ cot (exx) + C

B. tan (xex) + C

C. tan (ex) + C

D. cot (ex) + C

Let exx = t

Hence, the correct answer is B.

Let x3 = t

∴ 3x2 dx = dt

Let 2x = t

∴ 2dx = dt

Let 2 − t

⇒ −dx = dt

Let 5x = t

∴ 5dx = dt

Let x3 = t

∴ 3x2 dx = dt

#### Question 7:

From (1), we obtain

Let x3 = t

⇒ 3x2 dx = dt

Let tan x = t

∴ sec2x dx = dt

#### Question 11:

19×2+6x+5

∫19×2+6x+5dx=∫13x+12+22dx

Let (3x+1)=t

3 dx=dt

⇒∫13x+12+22dx=13∫1t2+22dt

=13×2tan-1t2+C

=16tan-13x+12+C

#### Question 16:

Equating the coefficients of x and constant term on both sides, we obtain

4A = 4 ⇒ A = 1

A + B = 1 ⇒ B = 0

Let 2x2 + x − 3 = t

∴ (4x + 1) dx dt

#### Question 17:

Equating the coefficients of x and constant term on both sides, we obtain

From (1), we obtain

From equation (2), we obtain

#### Question 18:

Equating the coefficient of x and constant term on both sides, we obtain

Substituting equations (2) and (3) in equation (1), we obtain

#### Question 19:

Equating the coefficients of x and constant term, we obtain

2A = 6 ⇒ A = 3

−9A + B = 7 ⇒ B = 34

∴ 6x + 7 = 3 (2x − 9) + 34

Substituting equations (2) and (3) in (1), we obtain

#### Question 20:

Equating the coefficients of x and constant term on both sides, we obtain

Using equations (2) and (3) in (1), we obtain

#### Question 21:

Let x2 + 2x +3 = t

⇒ (2x + 2) dx =dt

Using equations (2) and (3) in (1), we obtain

#### Question 22:

Equating the coefficients of x and constant term on both sides, we obtain

Substituting (2) and (3) in (1), we obtain

#### Question 23:

Equating the coefficients of x and constant term, we obtain

Using equations (2) and (3) in (1), we obtain

#### Question 24:

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

Hence, the correct answer is B.

#### Question 25:

equals

A.

B.

C.

D.

Hence, the correct answer is B.

#### Question 1:

Let

Equating the coefficients of x and constant term, we obtain

A + = 1

2A + B = 0

On solving, we obtain

A = −1 and B = 2

#### Question 2:

Let

Equating the coefficients of x and constant term, we obtain

A + B = 0

−3A + 3B = 1

On solving, we obtain

#### Question 3:

Let

Substituting x = 1, 2, and 3 respectively in equation (1), we obtain

A = 1, B = −5, and C = 4

#### Question 4:

Let

Substituting x = 1, 2, and 3 respectively in equation (1), we obtain

#### Question 5:

Let

Substituting x = −1 and −2 in equation (1), we obtain

A = −2 and B = 4

#### Question 6:

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (1 − x2) by x(1 − 2x), we obtain

Let

Substituting x = 0 and  in equation (1), we obtain

= 2 and B = 3

Substituting in equation (1), we obtain

#### Question 7:

Let

Equating the coefficients of x2x, and constant term, we obtain

A + C = 0

A + B = 1

B + C = 0

On solving these equations, we obtain

From equation (1), we obtain

#### Question 8:

Let

Substituting x = 1, we obtain

Equating the coefficients of x2 and constant term, we obtain

A + C = 0

−2A + 2B + C = 0

On solving, we obtain

#### Question 9:

Let

Substituting x = 1 in equation (1), we obtain

B = 4

Equating the coefficients of x2 and x, we obtain

A + C = 0

B − 2C = 3

On solving, we obtain

#### Question 10:

Let

Equating the coefficients of x2 and x, we obtain

#### Question 11:

Let

Substituting = −1, −2, and 2 respectively in equation (1), we obtain

#### Question 12:

It can be seen that the given integrand is not a proper fraction.

Therefore, on dividing (x3 + x + 1) by x2 − 1, we obtain

Let

Substituting = 1 and −1 in equation (1), we obtain

#### Question 13:

Equating the coefficient of x2x, and constant term, we obtain

A − B = 0

B − C = 0

A + C = 2

On solving these equations, we obtain

A = 1, B = 1, and C = 1

#### Question 14:

Equating the coefficient of x and constant term, we obtain

A = 3

2A + = −1 ⇒ B = −7

#### Question 15:

Equating the coefficient of x3x2, x, and constant term, we obtain

On solving these equations, we obtain

#### Question 16:

[Hint: multiply numerator and denominator by xn − 1 and put xn = t]

Multiplying numerator and denominator by x− 1, we obtain

Substituting t = 0, −1 in equation (1), we obtain

A = 1 and B = −1

#### Question 17:

[Hint: Put sin x = t]

Substituting t = 2 and then t = 1 in equation (1), we obtain

A = 1 and B = −1

#### Question 18:

Equating the coefficients of x3x2x, and constant term, we obtain

A + C = 0

B + D = 4

4A + 3C = 0

4B + 3D = 10

On solving these equations, we obtain

A = 0, B = −2, C = 0, and D = 6

#### Question 19:

Let x2 = t ⇒ 2x dx = dt

Substituting = −3 and = −1 in equation (1), we obtain

#### Question 20:

Multiplying numerator and denominator by x3, we obtain

Let x4 = t ⇒ 4x3dx = dt

Substituting t = 0 and 1 in (1), we obtain

A = −1 and B = 1

#### Question 21:

[Hint: Put ex = t]

Let ex = ⇒ ex dx = dt

Substituting t = 1 and t = 0 in equation (1), we obtain

A = −1 and B = 1

#### Question 22:

A.

B.

C.

D.

Substituting x = 1 and 2 in (1), we obtain

A = −1 and B = 2

Hence, the correct answer is B.

#### Question 23:

A.

B.

C.

D.

Equating the coefficients of x2x, and constant term, we obtain

A + B = 0

C = 0

A = 1

On solving these equations, we obtain

= 1, B = −1, and C = 0

Hence, the correct answer is A.

#### Question 1:

x sin x

Let I =

Taking x as first function and sin x as second function and integrating by parts, we obtain

#### Question 2:

Let I =

Taking x as first function and sin 3x as second function and integrating by parts, we obtain

#### Question 3:

Let

Taking x2 as first function and ex as second function and integrating by parts, we obtain

Again integrating by parts, we obtain

#### Question 4:

x logx

Let

Taking log x as first function and x as second function and integrating by parts, we obtain

#### Question 5:

x log 2x

Let

Taking log 2x as first function and x as second function and integrating by parts, we obtain

#### Question 6:

xlog x

Let

Taking log x as first function and x2 as second function and integrating by parts, we obtain

#### Question 7:

Let

Taking as first function and x as second function and integrating by parts, we obtain

#### Question 8:

Let

Taking  as first function and x as second function and integrating by parts, we obtain

#### Question 9:

Let

Taking cos−1 x as first function and x as second function and integrating by parts, we obtain

#### Question 10:

Let

Taking as first function and 1 as second function and integrating by parts, we obtain

#### Question 11:

Let

Taking  as first function and  as second function and integrating by parts, we obtain

#### Question 12:

Let

Taking x as first function and sec2x as second function and integrating by parts, we obtain

#### Question 13:

Let

Taking  as first function and 1 as second function and integrating by parts, we obtain

#### Question 14:

Taking  as first function and x as second function and integrating by parts, we obtain

I=log x 2∫xdx-∫ddxlog x 2∫xdxdx=x22log x 2-∫2log x .1x.x22dx=x22log x 2-∫xlog x dx

Again integrating by parts, we obtain

I = x22logx 2-log x ∫x dx-∫ddxlog x ∫x dxdx=x22logx 2-x22log x -∫1x.x22dx=x22logx 2-x22log x +12∫x dx=x22logx 2-x22log x +x24+C

#### Question 15:

Let

Let I = I1 + I2 … (1)

Where, and

Taking log x as first function and xas second function and integrating by parts, we obtain

Taking log x as first function and 1 as second function and integrating by parts, we obtain

Using equations (2) and (3) in (1), we obtain

#### Question 16:

Let

Let

⇒

∴

It is known that,

#### Question 17:

Let

Let  ⇒

It is known that,

#### Question 18:

Let⇒

It is known that,

From equation (1), we obtain

#### Question 19:

Also, let  ⇒

It is known that,

#### Question 20:

Let  ⇒

It is known that,

#### Question 21:

Let

Integrating by parts, we obtain

Again integrating by parts, we obtain

#### Question 22:

Let ⇒

= 2θ

⇒

Integrating by parts, we obtain

#### Question 23:

equals

Let

Also, let  ⇒

Hence, the correct answer is A.

#### Question 24:

equals

Let

Also, let  ⇒

It is known that,

Hence, the correct answer is B.

#### Question 10:

is equal to

A.

B.

C.

D.

Hence, the correct answer is A.

#### Question 11:

is equal to

A.

B.

C.

D.

Hence, the correct answer is D.

#### Question 1:

It is known that,

#### Question 2:

It is known that,

#### Question 3:

It is known that,

#### Question 4:

It is known that,

From equations (2) and (3), we obtain

#### Question 5:

It is known that,

#### Question 6:

It is known that,

#### Question 1:

By second fundamental theorem of calculus, we obtain

#### Question 2:

By second fundamental theorem of calculus, we obtain

#### Question 3:

By second fundamental theorem of calculus, we obtain

#### Question 4:

By second fundamental theorem of calculus, we obtain

#### Question 5:

By second fundamental theorem of calculus, we obtain

#### Question 6:

By second fundamental theorem of calculus, we obtain

#### Question 7:

By second fundamental theorem of calculus, we obtain

#### Question 8:

By second fundamental theorem of calculus, we obtain

#### Question 9:

By second fundamental theorem of calculus, we obtain

#### Question 10:

By second fundamental theorem of calculus, we obtain

#### Question 11:

By second fundamental theorem of calculus, we obtain

#### Question 12:

By second fundamental theorem of calculus, we obtain

#### Question 13:

By second fundamental theorem of calculus, we obtain

#### Question 14:

By second fundamental theorem of calculus, we obtain

#### Question 15:

By second fundamental theorem of calculus, we obtain

#### Question 16:

Let

Equating the coefficients of x and constant term, we obtain

A = 10 and B = −25

Substituting the value of I1 in (1), we obtain

#### Question 17:

By second fundamental theorem of calculus, we obtain

#### Question 18:

By second fundamental theorem of calculus, we obtain

#### Question 19:

By second fundamental theorem of calculus, we obtain

#### Question 20:

By second fundamental theorem of calculus, we obtain

#### Question 21:

equals

A.

B.

C.

D.

By second fundamental theorem of calculus, we obtain

Hence, the correct answer is D.

#### Question 22:

equals

A.

B.

C.

D.

By second fundamental theorem of calculus, we obtain

Hence, the correct answer is C.

#### Question 1:

When x = 0, t = 1 and when x = 1, t = 2

Also, let

#### Question 3:

Also, let x = tanθ ⇒ dx = sec2θ dθ

When x = 0, θ = 0 and when = 1,

Takingθas first function and sec2θ as second function and integrating by parts, we obtain

#### Question 4:

Let + 2 = t2 ⇒ dx = 2tdt

When x = 0,  and when = 2, = 2

#### Question 5:

Let cos x = t ⇒ −sinx dx = dt

When x = 0, = 1 and when

Let ⇒ dx = dt

#### Question 7:

Let x + 1 = ⇒ dx = dt

When x = −1, = 0 and when x = 1, = 2

#### Question 8:

Let 2x = t ⇒ 2dx = dt

When x = 1, t = 2 and when x = 2, t = 4

#### Question 9:

The value of the integral  is

A. 6

B. 0

C. 3

D. 4

Let cotθ = t ⇒ −cosec2θ dθdt

Hence, the correct answer is A.

#### Question 10:

If

A. cos x + x sin x

B. x sin x

C. x cos x

D. sin x cos x

Integrating by parts, we obtain

Hence, the correct answer is B.

#### Question 1:

Adding (1) and (2), we obtain

#### Question 2:

Adding (1) and (2), we obtain

#### Question 3:

Adding (1) and (2), we obtain

#### Question 4:

Adding (1) and (2), we obtain

#### Question 5:

It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

#### Question 6:

It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

#### Question 10:

Adding (1) and (2), we obtain

#### Question 11:

As sin(−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2is an even function.

It is known that if f(x) is an even function, then

#### Question 12:

Adding (1) and (2), we obtain

#### Question 13:

As sin(−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2is an odd function.

It is known that, if f(x) is an odd function, then

#### Question 14:

It is known that,

#### Question 15:

Adding (1) and (2), we obtain

#### Question 16:

Adding (1) and (2), we obtain

sin (π − x) = sin x

Adding (4) and (5), we obtain

Let 2x = t ⇒ 2dx = dt

When x = 0, = 0 and when

x=π2, t=π∴

I=12∫0πlog sin tdt-π2log 2

⇒I=I2-π2log 2       [from 3]

⇒I2=-π2log 2

⇒I=-πlog 2

#### Question 17:

It is known that,

Adding (1) and (2), we obtain

#### Question 18:

It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

#### Question 19:

Show that if f and g are defined as and

Adding (1) and (2), we obtain

#### Question 20:

The value of is

A. 0

B. 2

C. π

D. 1

It is known that if f(x) is an even function, then  and

if f(x) is an odd function, then

Hence, the correct answer is C.

#### Question 21:

The value of is

A. 2

B.

C. 0

D.

Adding (1) and (2), we obtain

Hence, the correct answer is C.

#### Question 1:

Equating the coefficients of x2x, and constant term, we obtain

A + B − C = 0

B + = 0

A = 1

On solving these equations, we obtain

From equation (1), we obtain

[Hint: Put]

#### Question 5:

On dividing, we obtain

#### Question 6:

Equating the coefficients of x2x, and constant term, we obtain

A + B = 0

C = 5

9A + = 0

On solving these equations, we obtain

From equation (1), we obtain

Let  a ⇒ dx = dt

#### Question 9:

Let sin x = t ⇒ cos x dx = dt

#### Question 12:

Let x= t ⇒ 4x3 dx = dt

#### Question 13:

Let ex = t ⇒ ex dx = dt

#### Question 14:

Equating the coefficients of x3x2x, and constant term, we obtain

A + C = 0

B + D = 0

4A + C = 0

4D = 1

On solving these equations, we obtain

From equation (1), we obtain

#### Question 15:

= cos3 x × sin x

Let cos x = t ⇒ −sin x dx = dt

#### Question 19:

Let I=∫sin-1x-cos-1xsin-1x+cos-1xdx

It is known that, sin-1x+cos-1x=π2

⇒I=∫π2-cos-1x-cos-1xπ2dx

=2π∫π2-2cos-1xdx

=2π.π2∫1.dx-4π∫cos-1xdx

=x-4π∫cos-1xdx           …(1)

Let I1=∫cos-1x dx

Also, let x=t⇒dx=2 t dt

⇒I1=2∫cos-1t.t dt

=2cos-1t.t22-∫-11-t2.t22dt

=t2cos-1t+∫t21-t2dt

=t2cos-1t-∫1-t2-11-t2dt

=t2cos-1t-∫1-t2dt+∫11-t2dt

=t2cos-1t-t21-t2-12sin-1t+sin-1t

=t2cos-1t-t21-t2+12sin-1t

From equation (1), we obtain

I=x-4πt2cos-1t-t21-t2+12sin-1t  =x-4πxcos-1x-x21-x+12sin-1x

=x-4πxπ2-sin-1x-x-x22+12sin-1x

#### Question 22:

Equating the coefficients of x2x,and constant term, we obtain

A + C = 1

3A + B + 2= 1

2A + 2B + C = 1

On solving these equations, we obtain

A = −2, B = 1, and C = 3

From equation (1), we obtain

#### Question 24:

Integrating by parts, we obtain

#### Question 26:

When = 0, = 0 and

When and when

#### Question 28:

When and when

As , therefore, is an even function.

It is known that if f(x) is an even function, then

#### Question 31:

From equation (1), we obtain

#### Question 32:

Adding (1) and (2), we obtain

#### Question 33:

From equations (1), (2), (3), and (4), we obtain

#### Question 34:

Equating the coefficients of x2x, and constant term, we obtain

A + C = 0

A + B = 0

B = 1

On solving these equations, we obtain

A = −1, C = 1, and B = 1

Hence, the given result is proved.

#### Question 35:

Integrating by parts, we obtain

Hence, the given result is proved.

#### Question 36:

Therefore, f (x) is an odd function.

It is known that if f(x) is an odd function, then

Hence, the given result is proved.

#### Question 37:

Hence, the given result is proved.

#### Question 38:

Hence, the given result is proved.

#### Question 39:

Integrating by parts, we obtain

Let 1 − x2 = t ⇒ −2x dx = dt

Hence, the given result is proved.

#### Question 40:

Evaluate as a limit of a sum.

It is known that,

#### Question 41:

is equal to

A.

B.

C.

D.

Hence, the correct answer is A.

#### Question 42:

is equal to

A.

B.

C.

D.

Hence, the correct answer is B.

#### Question 43:

If then is equal to

A.

B.

C.

D.

Hence, the correct answer is D.

#### Question 44:

The value of is

A. 1

B. 0

C. − 1

D.

Adding (1) and (2), we obtain

Hence, the correct answer is B.

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