Evaluate the following integrals:

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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