Evaluate the following integrals: (1-35) `int_0^pixsinx cos^2x dx`

Explanation

`I=int_(0)^pi frac{x sin x}{1+cos^2x}dx ---(1)`

Using `int_a^b f(x) dx=int_0^b f(a+b-x)dx`

`I=int_0^pi frac{(pi-x)sinx}{1+cos^2x}dx`

By adding (1) and (2) `=>2I=pi int_0^pi frac{sinx}{1+cos^2x}dx`

Let `cos x=t=>dt=-sinx dx`

`:. I=pi/2 int_(1)^(-1) frac{1}{1+t^2}(-dt)`

`=pi int_(0)^(1) frac{1}{1+t^2}dt`

`=pi tan^(-1)1`
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