Learn Miscellaneous Exercise 7 with Free Lessons & Tips

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Question 1:

If then is equal to

Solution 1:

Hence, the correct answer is D.

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Tasneem

Question 2:

Integrate the functions

Solution 2:

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

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Question 3:

Integrate the functions

Solution 3:

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Question 4:

Integrate the functions

[Hint: Put]

Solution 4:

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Question 5:

Integrate the functions

Solution 5:

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Question 6:

Integrate the functions

Solution 6:

On dividing, we obtain

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Question 7:

Integrate the functions

Solution 7:

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

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Question 8:

Integrate the functions

Solution 8:

Let x − a = t ⇒ dx = dt

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Question 9:

Integrate the functions

Solution 9:

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Question 10:

Integrate the functions

Solution 10:

Let sin x = t ⇒ cos x dx = dt

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Question 11:

Integrate the functions

Solution 11:

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Question 12:

Integrate the functions

Solution 12:

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Question 13:

Integrate the functions

Solution 13:

Let x⁴= t ⇒ 4x³dx = dt

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Question 14:

Integrate the functions

Solution 14:

Let e^x = t ⇒ e^xdx = dt

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Question 15:

Integrate the functions

Solution 15:

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

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Question 16:

Integrate the functions

Solution 16:

= cos³x × sin x

Let cos x = t ⇒ −sin x dx = dt

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Question 17:

Integrate the functions

Solution 17:

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Question 18:

Integrate the functions

Solution 18:

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Question 19:

Integrate the functions

Solution 19:

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Question 20:

Integrate the functions

Solution 20:

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Question 21:

Integrate the functions

Solution 21:

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Question 22:

Integrate the functions

Solution 22:

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

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Question 23:

Integrate the functions

Solution 23:

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Question 24:

Integrate the functions

Solution 24:

Integrating by parts, we obtain

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Question 25:

Evaluate the definite integrals

Solution 25:

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Question 26:

Evaluate the definite integrals

Solution 26:

When x = 0, t = 0 and

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Question 27:

Evaluate the definite integrals

Solution 27:

When and when

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Question 28:

Evaluate the definite integrals

Solution 28:

When and when

As , therefore, is an even function.

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

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Question 29:

Evaluate the definite integrals

Solution 29:

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Question 30:

Evaluate the definite integrals

Solution 30:

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Question 31:

Evaluate the definite integrals

Solution 31:

From equation (1), we obtain

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Question 32:

Evaluate the definite integrals

Solution 32:

Adding (1) and (2), we obtain

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Question 33:

Evaluate the definite integrals

Solution 33:

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

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Question 34:

Evaluate the definite integrals

Solution 34:

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.

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Question 35:

Evaluate the definite integrals

Solution 35:

Integrating by parts, we obtain

Hence, the given result is proved.

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Question 36:

Evaluate the definite integrals

Solution 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.

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Question 37:

Evaluate the definite integrals

Solution 37:

Hence, the given result is proved.

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Question 38:

Evaluate the definite integrals

Solution 38:

Hence, the given result is proved.

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Question 39:

Evaluate the definite integrals

Solution 39:

Integrating by parts, we obtain

Let 1 − x² = t ⇒ −2x dx = dt

Hence, the given result is proved.

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Question 40:

Evaluate as a limit of a sum.

Solution 40:

It is known that,

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Question 41:

is equal to

Solution 41:

Hence, the correct answer is A.

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Question 42:

is equal to

D. #new_question#

If then is equal to

Solution 42:

Hence, the correct answer is B.

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Question 43:

The value of is

A. 1

B. 0

C. − 1

Solution 43:

Adding (1) and (2), we obtain

Hence, the correct answer is B.

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Tasneem

All Exercises - Chapter 7 - Integrals

Miscellaneous Exercise 7

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Learn Miscellaneous Exercise 7 with Free Lessons & Tips

All Exercises - Chapter 7 - Integrals

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