Prove that , hence evaluate .

Let

Substitute t=a-x

⇒ dt=-dx.

Also when x=0, t=a and when x=a, t=0.

Hence,

Replacing t with x, we get

=RHS

Hence, .

Let

Using , we get

Adding (1) and (2), we get

Let cos x = t

⇒ -sin x dx=dt.

When x=0, t=1 and

when x=π, t=-1

Hence,