Question

Evaluate the following : `int_0^pi x/(1 + sin^2x)*dx`

Answer 1

Let I = `int_0^pi x/(1 + sin^2x)*dx`                     ...(1)

We use the property, `int_^a f(x)*dx = int_0^a f(a - x)*dx`

Here a = π.
Hence in I, changing x to π – x, we get

I = `int_0^pi (pi - x)/(1 + sin^2(pi - x))*dx`

= `int_0^pi (pi - x)/(1 + sin^2x)*dx` 

= `int_0^pi pi/(1 + sin^2x)*dx`

= `-int_0^(pi) x/(1 + sin^2x)*dx`

= `int_0^(pi) pi/(1 sin^2x)*dx - "I"`         ...[By (1)]

∴ 2I = `pi int_0^(pi) 1/(1 + sin^2x)*dx`
Dividing numerator and denominator by cos²x, we get

2I = `pi int_0^(pi) (sec^2x)/(sec^2x + tan^2x)*dx`

= `pi int_0^(pi) (sec^2x)/(1 + 2tan^2x)*dx`

Put tan x = t
∴ sec²x·dx = dt
When x = π, t = tan π = 0
When x = 0, t = tan 0 = 0

∴ 2I = `pi int_0^(pi) dt/(1 + 2^2)` = 0

∴ I = 0.                                    ...[∵ `int_a^a f(x)*dx = 0]`