`36/piint_(pi//6)^(pi//3)(dx)/(1+sqrt(cotx))` equals to

Explanation

`I=frac{36}{pi} int_{pi / 6}^{pi / 3} frac{dx}{1+sqrt{cot x}}=frac{36}{pi} int_{pi / 6}^{pi / 3} frac{sqrt{sin x} d x}{sqrt{sin x}+sqrt{cos x}} ldots(1)`


Using `int_{a}^{b} f(x) d x=int_{a}^{b} f(a+b-x) d x`


`I=frac{36}{pi} int_{pi / 6}^{pi / 3} frac{sqrt{sin (frac{pi}{2}-x)} dx}{sqrt{sin (frac{pi}{2}-x)}+sqrt{cos (frac{pi}{2}-x)}}=frac{36}{pi} int_{pi / 6}^{pi / 3} frac{sqrt{cos x} d x}{sqrt{cos x}+sqrt{sin x}} quad text { (2) } `

`2 I=frac{36}{pi} int_{pi / 6}^{pi / 3} dx=frac{36}{pi}[x]_{pi / 6}^{pi / 3}=6 Rightarrow I=3`