Evaluate: `int_0^pi(xtanx)/(s e c x+tanx)dx`

Explanation

`int_0^pi(xtanx)/(s e c x+tanx)dx`

`= int_0^pi frac{xfrac{sinx}{cosx}}{frac{1}{cosx}+frac{sinx}{cosx}}dx`

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

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

Adding 1 and 2

`2I=int_0^pi frac{pi sinx}{1+sinx}dx`

`=int_0^pi tanx secx dx-pi int_0^pi tan^2x dx`

`=pi int_0^pi secx tanxdx-pi v sec^xdx+piint_0^pi dx`
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