## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Miscellaneous Exercise

#### 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.

The value of is
A. 1
B. 0
C. − 1
D.