Integrations Micellineous exercise

Integrals

Question 1:

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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:

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Answer:

Question 3:

 [Hint: Put]

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Answer:

Question 4:

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Question 5:

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Answer:

On dividing, we obtain

Question 6:

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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:

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Answer:

Let x − a = t ⇒ dx = dt

Question 8:

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Answer:

Question 9:

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Answer:

Let sin x = t ⇒ cos x dx = dt

Question 10:

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Question 11:

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Question 12:

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Answer:

Let x⁴ = t ⇒ 4x³ dx = dt

Question 13:

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Answer:

Let e^x = t ⇒ e^x dx = dt

Question 14:

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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:

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Answer:

= cos³ x × sin x

Let cos x = t ⇒ −sin x dx = dt

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Question 16:

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Question 17:

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Question 18:

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Question 19:

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Answer:

From equation (1), we obtain

Question 20:

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Question 21:

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Answer:

Question 22:

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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:

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Question 24:

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Answer:

Integrating by parts, we obtain

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Question 25:

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Question 26:

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Answer:

When x = 0, t = 0 and 

Question 27:

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Answer:

When and when

Question 28:

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Answer:

When and when 

As , therefore, is an even function.

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

Question 29:

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Answer:

Question 30:

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Answer:

Question 31:

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Answer:

From equation (1), we obtain

Question 32:

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Answer:

Adding (1) and (2), we obtain

Question 33:

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Answer:

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

Question 34:

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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:

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Answer:

Integrating by parts, we obtain

Hence, the given result is proved.

Question 36:

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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:

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Answer:

Hence, the given result is proved.

Question 38:

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Answer:

Hence, the given result is proved.

uestion 39:

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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.

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Answer:

It is known that,

Question 41:

is equal to

A.

B.

C.

D.

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Answer:

Hence, the correct answer is A.

Question 42:

is equal to

A.

B.

C.

D.

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Answer:

Hence, the correct answer is B.

Question 43:

If then is equal to

A.

B.

C.

D.

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Answer:

Hence, the correct answer is D.

Question 44:

The value of is

A. 1

B. 0

C. − 1

D.

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Answer:

Adding (1) and (2), we obtain

Hence, the correct answer is B.