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Class 12 NCERT Solutions- Mathematics Part ii – Chapter 7– Integrals Exercise 7.10

Last Updated : 23 Jul, 2025

Class 12 NCERT Mathematics Part ii Chapter 7 Integrals Exercise 7.10 is about using the substitution method to evaluate the given integrals. The below article provides easy solutions for the questions in the exercise.

Evaluate Integrals 1 to 8 Using Substitution

Question 1

Solution:

Let

When and when

Question 2

Solution:

Let

Also, let

When and when

Question 3

Solution:

Let

Also, let

When , and when x=1,

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

Question 4

(Put x+2 = t2 )

Solution:

Letand

When and when

Question 5

Solution:

Let

When and when

Question 6

Solution:

Let

When and when ,

Question 7

Solution:

Let

When and when

Question 8.

Solution:

Let

When and when

Let

Then,

Choose Correct Answer in Question 9 and 10

Question 9.

Value of the integral is:

  • [A] 6
  • [B] 0
  • [C] 3
  • [D] 4

Correct Answer is Option [A] 6

Solution:

Let I =

= [x3(x-2 -1)]1/3/x4 dx

= x(x-2 - 1)1/3/x4 dx

= (x-2 - 1)1/3x-3 dx...(i)

let, x-2 - 1 = t

-2x-3 = dt/dx

x-3dx = -1/2dt

Changing limit of Integration

when, x = 1/3, t = x-2 -1 = (1/3)-2 - 1 = 8

when, x = 1, t = x-2 -1 = (1)-2 - 1 = 0

From (i)

I = -1/2 t1/3.dt

I = -1/2(t4/3/{4/3})08

I = -1/2.3/4{0 - 16}

I = 6

Question 10.

If , then f'(x) is

  • [A] cosx + x sinx
  • [B] x sinx
  • [C] x cosx
  • [D] sinx + x cosz

Correct Answer is Option [B] x sinx

Solution:

f(x) = ...(i)

let,

  • u = t
  • v = cos t

du = dt

dv = -sint.dt

using, ∫udv = uv − ∫vdu

from eq(i)

= [-tcos(t)]x0 -

= [-x.cosx - (-0.cos0)} -

= -x.cosx +

= -x.cos(x) + [sin (t)]x0

= -x.cos(x) + sin(x) - sin (0)

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