Class 12 RD Sharma Solutions- Chapter 20 Definite Integrals - Exercise 20.4 Part A

Evaluate each of the following integrals (1-16):

Question 1.

Solution:

We know that 

so,

we know, 

if 

I = 

then 

I =

2I = 

l=π 

Question 2.

Solution:

We know that 

So, 

if 

I = 

then 

I = 

2I = 

2I= 

2I = 

2I=0

I=0

Question 3.

Solution:

We know 

So,

if

I=

then 

I=

So

2I=

2I = 

2I=

2I=π/6

I=π/12

Question 4. 

Solution:

We know 

So,

if 

I = 

then,

 I = 

2I = 

2I = 

2I = 

2I=π/6

I=π/12

Question 5. 

Solution:

We know 

so,

if 

then,

we know if 

f(x) is even 

f(x) is odd 

Here, f(x) = tan²x which is even

hence,

I = 

Question 6. 

Solution:

We know 

So, 

if, then 

So, 

Question 7. 

Solution:

We know 

Hence, 

if, 

then 

so,

Question 8. 

Solution:

We know 

hence, 

if 

Then, 

So, 

Question 9.  

Solution:

if f(x) is even 

if f(x) is odd 

here,   is odd and  

is even

Hence,

2

Question 10.  

Solution:

if 

then, 

Question 11.  

Solution:

let, 

we know that, 

hence,

Question 12.  

Solution:

Let, 

we know that ,

so, 

then, 

Question 13.  

Solution:

We know that, 

So,

then,

I

Question 14.  

Solution:

We know that, 

so,

then,

Question 15.  

Solution:

We know that, 

Let, 

hence,

Question 16. If f(a+b-x)=f(x), then prove that 

Solution:

Summary

This exercise typically focuses on evaluating definite integrals using various methods and properties. Key concepts often covered include:

• Basic definite integral evaluation
• Properties of definite integrals
• Integration by substitution
• Integration by parts
• Evaluating integrals with trigonometric functions
• Dealing with even and odd functions in definite integrals