Evaluate the following integrals:

Assume $\cot x=t$

$\Rightarrow \mathrm{d}(\cot \mathrm{x})=\mathrm{dt}$

$\Rightarrow-\operatorname{cosec}^{2} \mathrm{x} \cdot \mathrm{d} \mathrm{x}=\mathrm{dt}$

$\Rightarrow \mathrm{dt}=\frac{-\mathrm{dt}}{\csc ^{2} \mathrm{x}}$

$\therefore$ Substituting $\mathrm{t}$ and $\mathrm{dt}$ in the given equation we get

$\Rightarrow \int \mathrm{t}^{3} \csc ^{2} \mathrm{x} \cdot \frac{-\mathrm{dt}}{\csc ^{2} \mathrm{x}}$

$\Rightarrow \int-\mathrm{t}^{3} \cdot \mathrm{dt}$

$\Rightarrow-\int \mathrm{t}^{3} \cdot \mathrm{dt}$

$\Rightarrow \frac{-t^{4}}{4}+c$

But $t=\cot x$

$\Rightarrow \frac{-\cot ^{4} x}{4}+c$