Evaluate the following integral: `int_(pi//6)^(pi//2)(cos e c\ xcotx)/(1+cos e c^2x)dx`

Explanation

Let `I=int_{pi / 6}^{pi / 2} frac{{cosec} x cot x}{1+{cosec}^{2} x} d x`


Let `sin x=t`


`cos x d x=d t`


So, when `x=frac{pi}{6}, t=frac{1}{2}` and when `x=frac{pi}{2}, {t}=1.`


`I=int_{frac{1}{2}}^{1} frac{1}{1+t^{2}} d t`

`=tan ^{-1} t |[(1), (frac{1}{2})]`

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