Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

Answer:

Equating the coefficients of x², x, and constant term, we obtain

−A + B − C = 0

B + C = 0

A = 1

On solving these equations, we obtain

From equation (1), we obtain

Question 2:

Answer:

Question 3:

 [Hint: Put ]

Answer:

Question 4:

Answer:

Question 5:

Answer:

On dividing, we obtain

Question 6:

Answer:

Equating the coefficients of x², x, and constant term, we obtain

A + B = 0

B + C = 5

9A + C = 0

On solving these equations, we obtain

From equation (1), we obtain

Question 7:

Answer:

Let x − a = t ⇒ dx = dt

Question 8:

Answer:

Question 9:

Answer:

Let sin x = t ⇒ cos x dx = dt

Question 10:

Answer:

Question 11:

Answer:

Question 12:

Answer:

Let x⁴ = t ⇒ 4x³ dx = dt

Question 13:

Answer:

Let ex = t ⇒ ex dx = dt

Question 14:

Answer:

Equating the coefficients of x³, x², x, and constant term, we obtain

A + C = 0

B + D = 0

4A + C = 0

4B + D = 1

On solving these equations, we obtain

From equation (1), we obtain

Question 15:

Answer:

= cos³ x × sin x

Let cos x = t ⇒ −sin x dx = dt

Question 16:

Answer:

Question 17:

Answer:

Question 18:

Answer:

Question 19:

Question 20:

Answer:

Question 21:

Answer:

Question 22:

Answer:

Equating the coefficients of x², x,and constant term, we obtain

A + C = 1

3A + B + 2C = 1

2A + 2B + C = 1

On solving these equations, we obtain

A = −2, B = 1, and C = 3

From equation (1), we obtain

Question 23:

Answer:

Question 24:

Answer:

Integrating by parts, we obtain

Question 25:

Answer:

Question 26:

Answer:

When x = 0, t = 0 and 

Question 27:

Answer:

When  and when

Question 28:

Answer:

When  and when 

As  , therefore,  is an even function.

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

Question 29:

Answer:

Question 30:

Answer:

Question 31:

Answer:

From equation (1), we obtain

Question 32:

Answer:

Adding (1) and (2), we obtain

Question 33:

Answer:

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

Question 34:

Answer:

Equating the coefficients of x², x, 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:

Answer:

Integrating by parts, we obtain

Hence, the given result is proved.

Question 36:

Answer:

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:

Answer:

Hence, the given result is proved.

Question 38:

Answer:

Hence, the given result is proved.

Question 39:

Answer:

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.

Answer:

It is known that,

Question 41:

is equal to

A. 

B. 

C. 

D. 

Answer:

Hence, the correct Answer is A.

Question 42:

is equal to

A. 

B. 

C. 

D. 

Answer:

Hence, the correct Answer is B.

Question 43:

If  then  is equal to

A. 

B. 

C. 

D. 

Answer:

Hence, the correct Answer is D.

Question 44:

The value of  is

A. 1

B. 0

C. − 1

D. 

Answer:

Adding (1) and (2), we obtain

Hence, the correct Answer is B.