Evaluate:

Let

…(i)

We know that,

So, if

[∵, sin (π – x) = sin x]

…(ii)

Adding these two I’s from equations (i) and (ii), we get

Solving further,

Now, we can easily integrate the Right Hand Side (RHS) of the equation.

Let us divide the numerator and denominator by cos x.

…(iii)

Now,

Let t = sec x + tan x.

Differentiating both sides with respect to x, we get

Again put t in place of tan x + sec x.

⇒ dt = t.sec x dx

Putting sec x + tan x = t and in equation (iii), we get

Substituting t = sec x + tan x, we get

Putting the upper and lower limits,

⇒ I = π