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# Class 12 RD Sharma Solutions – Chapter 19 Indefinite Integrals – Exercise 19.5

• Last Updated : 07 Apr, 2021

### Question 1.

Solution:

Given integral,

On Multiplying and dividing with 2, we get

â‡’

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

We get

â‡’

â‡’

â‡’

â‡’

### Question 2.

Solution:

Given integral,

Let x + 2 =t â‡’ x = t – 2

On differentiating on both sides,

dx = dt

On substituting it in given integral, we get

â‡’

â‡’

We know that,              [where c is any arbitrary constant]

â‡’

â‡’

Replacing x in terms of t

â‡’

â‡’

### Question 3.

Solution:

Given integral,

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

We get

â‡’

â‡’

â‡’

### Question 4.

Solution:

Given integral,

Let 3x + 5 = t

â‡’ x = (t – 5)/3

On differentiating both sides,

dx = dt/3

On replacing the x terms with t,

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

We get

â‡’

â‡’

On replacing t with x terms

â‡’

â‡’

â‡’

### Question 5.

Solution:

Given integral,

On multiplying and dividing it with 3

â‡’

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

We get

â‡’

â‡’

â‡’

â‡’

â‡’

### Question 6.

Solution:

Given integral,

Let 7x + 9 = t

â‡’ x = (t – 9)/7

On differentiating both sides,

dx = dt/7

On replacing x terms with t

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

â‡’

â‡’

On replacing t with x terms

â‡’

â‡’

### Question 7.

Solution:

Given integral,

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

â‡’

â‡’

â‡’

### Question 8.

Solution:

Given integral,

Let 1 + 3x = t

â‡’ x = (t – 1)/3

On differentiating both sides, we get

dx = dt/3

On replacing x with t

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

â‡’

Now on replacing t in terms of x

â‡’

â‡’

â‡’

### Question 9.

Solution:

Given integral,

Let 2x – 1 = t2

â‡’ x = (t2 + 1)/2

On differentiating on both sides,

dx = tdt

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

â‡’

On replacing t with x terms

â‡’

â‡’

â‡’

â‡’

### Question 10.

Solution:

Given integral,

On multiplying and dividing the given integral with

We know that (a + b)(a – b) = a2 – b2

â‡’

â‡’

â‡’

â‡’

By using the formula,

[where c is any arbitrary constant]

â‡’

â‡’

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