Evaluate the following integrals: (1-35) `int_0^(pi//2)(sin^n x)/(sin^n+cos^n x)dx`

Explanation

`I=int_0^(pi/2) frac{sin^nx}{sin^nx+cos^nx}dx ---(i)`

`=int_0^(pi/2) frac{sin^n(pi/2-x)}{sin^n(pi/2-x)+cos^n(pi/2-x)}dx`

`=int_0^(pi/2) frac{cos^n}{sin^nx+cos^nx} ---(ii)`

`2I=int_0^(pi/2) frac{sin^nx}{sin^nx+cos^nx}dx+int_0^(pi/2) frac{cos^n}{sin^nx+cos^nx}`

`2I=int_0^(pi/2)1dx`

`2I=pi/2-0`

`I=pi/4`