# Chapter 7:Miscellaneous Exercise

Integrate the functions in Exercises 1 to 24.(Chapter 7 Miscellaneous integration ncert maths solution class 12)

Question 1:

Solution:

Let

Let

Let

Let

From equation (1), we have

Now,

Question 2:

Solution: let

Question 3:

Solution:

Let

Question 4:

Solution: Let

Question 5:

Solution: Let

Let

Dividing by denominator , we have

Question 6:

Solution: Let

Let

Let

Let

From equation (1), we have

Question 7:

Solution: Let

Let

Question 8:

Solution: Let

Question 9:

Solution: Let

Let

Question 10:

Solution: Let

Question 11:

Solution : Let

Multiplying and dividing by , we have

Question 12:

Solution: Let

Let

Question 13:

Solution: Let

Let

Since,

Let

Again Let

Putting the value of A and B in (i) and integrate

Question 14:

Solution: Let

Let

Let

Let

Putting the value of A and b and integrate

Question 15:

Solution : Let

Let

Question 16:

Solution : Let

Let

Question 17:

Solution : Let

Let

Question 18:

Solution : Let

Let

Question 19:

Solution : Let

We know that:

Therefore,

Let and let

From equation (1), we have

Question 20:

Solution : Let

Let

Question 21:

Solution : Let

We know that:

Therefore,

Question 22:

Solution: Let

Let

Let

Let

From equation (1), we have

Therefore,

Question 23:

Solution :Let

Let

Question 24:

Solution: Let

Let

Therefore,

Integrating by parts, we have

Evaluate the definite integrals in Exercises 25 to 33.

Question 25:

Solution: Let

Let

Therefore,

Question 26:

Solution: Let

Divide by in numerator and denominator

Let

When and when

Question 27:

Solution: Let

Let

When and when

Question 28:

Solution: Let

Let

Again taking

Squaring both side

When

and When

Now

Question 29:

Solution: Let

Question 30:

Solution: Let

Let

Again taking

Squaring both side

When

and When

Now

Question 31:

Solution: Let

Let

When and when

Question 32:

Solution: Let

Question 33:

Solution: Let

Prove the following (Exercises 34 to 39)

Question 34:

Solution: Let

Let

Let

Let

respectively.

Putting the value in eq (i) and integrate

Hence proved.

Question 35:

Solution: Let

Hence proved.

Question 36:

Solution: Let

Now, consider

is an odd function and hence it is clearly known to us that when f(x) is an odd function, then

Hence proved.

Question 37:

Solution: Let

Let

Hence proved.

Queston 38:

Solution: Let

Let

When and when

Hence proved

Question 39:

Solution: Let

Let

when and when

Hence proved.

Question 40: Evaluate as a limit of sum.

Solution: Let

where,

In this case,

Choose the correct answers in Exercises 41 to 44.

Question 41: is equal to

A.

B.

C.

D.

Solution: The correct answer is option (A)

Let .

Let

, where C is any arbitrary constant.

Hence, the correct answer is option (A).

Question 42: is equal to

A.

B.

C.

D.

Solution : The correct answer is option (B)

Let

Let

where C is any arbitrary constant.

Hence, the correct answer is option (B).

Question 43: If is equal to

A.

B.

C.

D.

Solution: The correct answer is option (D)

Let

From (i) and (ii)

Hence, the correct answer is option (D).

Question 44: The value of is equal to

A.

B.

C.

D.

Solution : The correct answer is option (D).

Let

Hence, the correct answer is option (D).

gmath.in