Question:
Evaluate the following integrals:
$\int \cot ^{n} x \operatorname{cosec}^{2} x d x, n \neq-1$
Solution:
Let I $=\int \cot ^{n} x \operatorname{cosec}^{2} x d x$
Let $\cot x=t \Rightarrow-\operatorname{cosec}^{2} x d x=d t$
$\Rightarrow I=-\int t^{n} d t$
$\Rightarrow I=-\frac{t^{n+1}}{n+1}+c$
$\Rightarrow I=-\frac{\cot ^{n+1} x}{n+1}+c$
Therefore, $\int \cot ^{n} x \operatorname{cosec}^{2} x d x=-\frac{\cot ^{n+1} x}{n+1}+c$
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