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

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

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Question 1:

Answer:

Let 

Equating the coefficients of x and constant term, we obtain

A + B = 1

2A + 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

−3A + 3B = 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 − x²) by x(1 − 2x), 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 x², 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 x² and constant term, we obtain

A + C = 0

−2A + 2B + C = 0

On solving, we obtain

Question 9:

Answer:

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:

Answer:

Let 

Equating the coefficients of x² 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 (x³ + x + 1) by x² − 1, we obtain

Let 

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

Question 13:

Answer:

Equating the coefficient of x², 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

2A + B = −1 ⇒ B = −7

Question 15:

Answer:

Equating the coefficient of x³, x², 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 xⁿ = 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

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Question 18:

Answer:

Equating the coefficients of x³, x², x, 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:

Answer:

Let x² = t ⇒ 2x dx = dt

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

Question 20:

Answer:

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 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 x², 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.