Question

`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.

Options

• 0

• `π/4`

• `π^2/4`

• `π^2/2`

Answer 1

`int_0^π(xsinx)/(1 + cos^2x)dx` equals `underlinebb(π^2/4)`.

Explanation:

I = `int_0^π (xsinx)/(1 + cos^2x)dx` ...(1)

I = `int_0^π ((π - x)sinx)/(1 + cos^2x)dx`  ...(2)

Adding, 2I = `int_0^π (πsinx)/(1 + cos^2x)dx`

I = `π/2int_-1^1 (dt)/(1 + t^2) [cosx = t]` ...(sinx dx = dt)

I = `π/2(tan^-1 t)_-1^1`

= `π/2.π/2`

= `π^2/4`