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Learn Exercise 7.5 with Free Lessons & Tips

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Question 1:

[Hint: Put e^x = t]

Solution 1:

Let e^x = t ⇒ e^xdx = dt

Substituting t = 1 and t = 0 in equation (1), we obtain

A = −1 and B = 1

Comments

Tasneem

Question 2:

Integrate the rational functions

Solution 2:

Let

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

Comments

Shivam Sharma

Question 3:

Integrate the rational functions

Solution 3:

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

Comments

Tasneem

Question 4:

Integrate the rational functions

Solution 4:

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

Comments

Shivam Sharma

Question 5:

Integrate the rational functions

Solution 5:

Let

Equating the coefficients of x and constant term, we obtain

A + B = 0

−3A + 3B = 1

On solving, we obtain

Comments

Debesh Mukhopadhyay

Question 6:

Integrate the rational functions

Solution 6:

Let

On solving

A = 1, B = -5, C = 4

Comments

Naved Ali

Question 7:

Integrate the rational functions

Solution 7:

Let

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

A = −2 and B = 4

Comments

Shivam Sharma

Question 8:

Integrate the rational functions

Solution 8:

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

Comments

Shivam Sharma

Question 9:

Integrate the rational functions

Solution 9:

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

Comments

Shivam Sharma

Question 10:

Integrate the rational functions

Solution 10:

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

Comments

Shivam Sharma

Question 11:

Integrate the rational functions

Solution 11:

Let

Equating the coefficients of x² and x, we obtain

Comments

Tasneem

Question 12:

Integrate the rational functions

Solution 12:

Let

Substituting x = −1, −2, and 2 respectively in equation (1), we obtain

Comments

Shivam Sharma

Question 13:

Integrate the rational functions

Solution 13:

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

Comments

Shivam Sharma

Question 14:

Integrate the rational functions

Solution 14:

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

Comments

Shivam Sharma

Question 15:

Integrate the rational functions

Solution 15:

Equating the coefficient of x and constant term, we obtain

A = 3

2A + B = −1 ⇒ B = −7

Comments

Shivam Sharma

Question 16:

Integrate the rational functions

Solution 16:

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

On solving these equations, we obtain

Comments

Tasneem

Question 17:

Solution 17:

Substituting x = 1 and 2 in (1), we obtain

A = −1 and B = 2

Hence, the correct answer is B.

Comments

Shivam Sharma

Question 18:

Solution 18:

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.

Comments

Tasneem

All Exercises - Chapter 7 - Integrals

Exercise 7.5

Miscellaneous Exercise 7

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Learn Exercise 7.5 with Free Lessons & Tips

All Exercises - Chapter 7 - Integrals

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