Evaluate the following integrals:

Let I $=\int \cot ^{5} x d x$

$\Rightarrow I=\int \cot ^{2} x \cot ^{3} x d x$

$\Rightarrow I=\int\left(\operatorname{cosec}^{2} x-1\right) \cot ^{3} x d x$

$\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d x-\int \cot ^{3} x d x$

$\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d x-\int\left(\operatorname{cosec}^{2} x-1\right) \cot x d x$

$\Rightarrow I=\int \cot ^{3} x \operatorname{cosec}^{2} x d x-\int\left(\operatorname{cosec}^{2} x \cot x\right) d x+\int \cot x d x$

Let $\cot x=t$, then

$\Rightarrow-\operatorname{cosec}^{2} x d x=d t$

$\Rightarrow I=-\int t^{3} d t+\int t d t+\int \cot x d x$

$\Rightarrow I=-\frac{t^{4}}{4}+\frac{t^{2}}{2}+\log |\sin x|+c$

$\Rightarrow I=-\frac{\cot ^{4} x}{4}+\frac{\cot ^{2} x}{2}+\log |\sin x|+c$

Therefore, $\int \cot ^{5} x d x=-\frac{\cot ^{4} x}{4}+\frac{\cot ^{2} x}{2}+\log |\sin x|+c$