What is  equal to?

Concept:

If f(x)  is even function then f(-x) = f(x)

If f(x)  is odd function then f(-x) = -f(x)

Properties of definite integral

If f(x)  is even function then 

If f(x)  is odd function then 

Calculation:

Let I = 

= I₁ + I₂

Now, 

I₁ 

Here f(x) = x²

Replace x by -x, we get

⇒ f(-x) = (-x)² = x²

⇒ f(-x) = f(x)

So, f(x) is even function.

As we know, If f(x) even function then 

Therefore, I₁ = 

Now, 

I₂ = 

Here f(x) = sin x

Replace x by -x, we get

⇒ f(-x) = sin (-x) = -sin x                (∵ sin (-θ) = - sin θ)

⇒ f(-x) = -f(x)

So, f(x) is odd function.

As we know, If f(x) even function then 

I = I₁ + I₂ =  + 0 =