Question

Evaluate:

`int_0^(π/2) (sin 2x)/(1 + sin^4x)dx`

Answer 1

Let I = `int_0^(π/2) (sin 2x)/(1 + sin^4x)dx`

Put sin²x = t

`\implies` 2 sin x cos x dx = dt

i.e. sin 2x dx = dt  ...(∵ sin 2θ = 2 sin θ cos θ)

When x = 0, t = 0 and When x = `π/2`, t = 1

∴ I = `int_0^1 (dt)/(1 + t^2)`

= `[tan^-1 t]_0^1`

= tan^–1 1 – tan^–1 0

= `π/4 - 0`

= `π/4`