Evaluate the following integral: `int_0^(pi//2)(sin^(3//2)\ )/(sin^(3//2)x+cos^(3//2)x)dx`

Explanation

`int_0^(pi//2)(sin^(3//2)\ )/(sin^(3//2)x+cos^(3//2)x)dx --(1)`

`=int_0^(pi//2)frac{sin^(3/2)(pi/2 -x)}{sin^(3/2)(pi/2 -x)+cos^(3/2)(pi/2 -x)}dx`

`I=int_0^(pi//2) frac{cos^(3/2)x}{cos^(3/2)x+sin^(3/2)x}dx --(2)`

Addng 1 and 2,

`2I=int_0^(pi//2) frac{sin^(3//2)x+cos^(3//2)x}{sin^(3//2)x+cos^(3//2)x}dx`

`2I=pi/2`

`I=pi/4`