## NCERT Solutions for Class 12 Maths Chapter 7 – Integrals Ex 7.5

#### Page No 322:

#### Question 1:

#### Answer:

Let

Equating the coefficients of

*x*and constant term, we obtain*A*+

*B*= 1

2

*A*+*B*= 0
On solving, we obtain

*A*= −1 and

*B*= 2

#### Question 2:

#### Answer:

Let

Equating the coefficients of

*x*and constant term, we obtain*A*+

*B*= 0

−3

*A*+ 3*B*= 1
On solving, we obtain

#### Question 3:

#### Answer:

Let

Substituting

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

*B*= −5, and

*C*= 4

#### Question 4:

#### Answer:

Let

Substituting

*x*= 1, 2, and 3 respectively in equation (1), we obtain#### Question 5:

#### Answer:

Let

Substituting

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

*B*= 4

#### Question 6:

#### Answer:

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

Therefore, on dividing (1 −

*x*^{2}) by*x*(1 − 2*x*), we obtain
Let

Substituting

*x*= 0 and in equation (1), we obtain*A*= 2 and

*B*= 3

Substituting in equation (1), we obtain

#### Question 7:

#### Answer:

Let

Equating the coefficients of

*x*^{2},*x*, 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:

#### Answer:

Let

Substituting

*x*= 1, we obtain
Equating the coefficients of

*x*^{2}and constant term, we obtain*A*+

*C*= 0

−2

*A*+ 2*B*+*C*= 0
On solving, we obtain

#### Question 9:

#### Answer:

Let

Substituting

*x*= 1 in equation (1), we obtain*B*= 4

Equating the coefficients of

*x*^{2}and*x*, we obtain*A*+

*C*= 0

*B*− 2

*C*= 3

On solving, we obtain

#### Question 10:

#### Answer:

Let

Equating the coefficients of

*x*^{2}and*x*, we obtain#### Question 11:

#### Answer:

Let

Substituting

*x*= −1, −2, and 2 respectively in equation (1), we obtain#### Question 12:

#### Answer:

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

Therefore, on dividing (

*x*^{3}+*x*+ 1) by*x*^{2}− 1, we obtain
Let

Substituting

*x*= 1 and −1 in equation (1), we obtain#### Question 13:

#### Answer:

Equating the coefficient of

*x*^{2},*x*, 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:

#### Answer:

Equating the coefficient of

*x*and constant term, we obtain*A*= 3

2

*A*+*B*= −1 ⇒*B*= −7#### Question 15:

#### Answer:

Equating the coefficient of

*x*^{3},*x*^{2},*x*, and constant term, we obtain
On solving these equations, we obtain

#### Question 16:

[Hint: multiply numerator and denominator by

*x*^{n}^{ − 1}and put*x*^{n}=*t*]#### Answer:

Multiplying numerator and denominator by

*x*^{n }^{− 1}, we obtain
Substituting

*t*= 0, −1 in equation (1), we obtain*A*= 1 and

*B*= −1

#### Question 17:

[Hint: Put sin

*x*=*t*]#### Answer:

Substituting

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

*B*= −1

#### Page No 323:

#### Question 18:

#### Answer:

Equating the coefficients of

*x*^{3},*x*^{2},*x*, and constant term, we obtain*A*+

*C*= 0

*B*+

*D*= 4

4

*A*+ 3*C*= 0
4

*B*+ 3*D*= 10
On solving these equations, we obtain

*A*= 0,

*B*= −2,

*C*= 0, and

*D*= 6

#### Question 19:

#### Answer:

Let

*x*^{2}=*t*⇒ 2*x**dx*=*dt*
Substituting

*t*= −3 and*t*= −1 in equation (1), we obtain#### Question 20:

#### Answer:

Multiplying numerator and denominator by

*x*^{3}, we obtain
Let

*x*^{4}=*t*⇒ 4*x*^{3}*dx*=*dt*
Substituting

*t*= 0 and 1 in (1), we obtain*A*= −1 and

*B*= 1

#### Question 21:

[Hint: Put

*e*^{x}=*t*]#### Answer:

Let

*e*^{x}=*t*⇒*e*^{x}*dx*=*dt*
Substituting

*t*= 1 and*t*= 0 in equation (1), we obtain*A*= −1 and

*B*= 1

#### Question 22:

**A.**

**B.**

**C.**

**D.**

#### Answer:

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.**

#### Answer:

Equating the coefficients of

*x*^{2},*x*, and constant term, we obtain*A*+

*B*= 0

*C*= 0

*A*= 1

On solving these equations, we obtain

*A*= 1,

*B*= −1, and

*C*= 0

Hence, the correct answer is A.

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