Class 12 RD Sharma Solutions - Chapter 19 Indefinite Integrals - Exercise 19.5

Question 1.  

Solution:

Given integral, 

On Multiplying and dividing with 2, we get

⇒  

⇒ 

⇒ 

⇒ 

By using the formula,

     [where c is any arbitrary constant]

We get

⇒ 

⇒ 

⇒ 

⇒      

Question 2. 

Solution:

Given integral, 

Let x + 2 =t ⇒ x = t - 2

On differentiating on both sides, 

dx = dt

On substituting it in given integral, we get

⇒ 

⇒ 

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

⇒ 

⇒ 

Replacing x in terms of t

⇒ 

⇒ 

Question 3. 

Solution:

Given integral, 

⇒ 

⇒ 

⇒ 

By using the formula, 

        [where c is any arbitrary constant]

We get

⇒ 

⇒ 

⇒ 

Question 4. 

Solution: 

Given integral, 

Let 3x + 5 = t

⇒ x = (t - 5)/3

On differentiating both sides, 

dx = dt/3

On replacing the x terms with t,

⇒ 

⇒ 

⇒ 

By using the formula,

         [where c is any arbitrary constant]

We get

⇒ 

⇒ 

On replacing t with x terms

⇒ 

⇒ 

⇒ 

Question 5. 

Solution:

Given integral, 

On multiplying and dividing it with 3

⇒ 

⇒ 

⇒ 

⇒ 

By using the formula,

        [where c is any arbitrary constant]

We get

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Question 6. 

Solution:

Given integral, 

Let 7x + 9 = t

⇒ x = (t - 9)/7

On differentiating both sides, 

dx = dt/7

On replacing x terms with t

⇒ 

⇒ 

⇒ 

By using the formula,

        [where c is any arbitrary constant]

⇒ 

⇒ 

On replacing t with x terms

⇒ 

⇒   

Question 7. 

Solution:

Given integral, 

⇒ 

⇒ 

By using the formula,

             [where c is any arbitrary constant]

⇒ 

⇒ 

⇒   

Question 8. 

Solution:

Given integral, 

Let 1 + 3x = t

⇒ x = (t - 1)/3

On differentiating both sides, we get 

dx = dt/3

On replacing x with t

⇒ 

⇒ 

⇒ 

By using the formula,

         [where c is any arbitrary constant]

⇒ 

Now on replacing t in terms of x

⇒ 

⇒ 

⇒   

Question 9. 

Solution:

Given integral, 

Let 2x - 1 = t²

 ⇒ x = (t² + 1)/2

On differentiating on both sides, 

dx = tdt 

 ⇒ 

⇒ 

⇒ 

By using the formula,

            [where c is any arbitrary constant]

⇒ 

On replacing t with x terms

⇒ 

⇒ 

⇒ 

⇒   

Question 10. 

Solution:

Given integral, 

On multiplying and dividing the given integral with 

We know that (a + b)(a - b) = a² - b²

⇒ 

⇒ 

⇒ 

⇒ 

By using the formula,

           [where c is any arbitrary constant]

⇒ 

⇒