Evaluate the following integrals:
Question:

Evaluate $\int \operatorname{cosec}^{4} 2 x d x$

Solution:

$I=\int \operatorname{cosec}^{4} 2 x d x$

$=\int \operatorname{cosec}^{2} 2 x\left(1+\cot ^{2} 2 x\right) d x$

$=\int \operatorname{cosec}^{2} 2 x d x+\int \operatorname{cosec}^{2} 2 x \cot ^{2} 2 x d x$

Let us consider that $\cot 2 x=t$ then $-2 \cdot \operatorname{cosec}^{2} 2 x d x=d t$

$I=-\frac{\cot (2 x)}{2}-\frac{1}{2} \cdot\left(t^{2} d t\right)$

$I=-\frac{\cot (2 x)}{2}-\frac{1}{6} \cdot(\cot 2 x)^{3}$