Question

Evaluate $\int {\sec }^{2}x{\csc }^{2}xdx$.

Answer 1

$\int {\sec }^{2}x{\csc }^{2}xdx$

$=\int \frac{dx}{{\cos }^{2}x{\sin }^{2}x}$

$=\int \frac{4dx}{4{\cos }^{2}x{\sin }^{2}x}$

$=\int \frac{4dx}{{\left(2\sin x\cos x\right)}^2}$

$=\int \frac{4dx}{{\left(\sin 2x\right)}^2}$

$=\int 4{\csc }^{2}2xdx$

Let $u=\cot 2x\Rightarrow du=-2{\csc }^{2}2xdx$

$=-4\int \frac{du}{2}$

$=-2u+c$ where $c$ is the constant of integration.

$=-2\cot 2x+c$