Question

Find m if $\int {\sec }^{2}x(1-{\cot }^{2}x)dx$ = $m\text{cosec}\left(2x\right)+C$

Answer 1

$\int {\sec }^{2}x(1-{\cot }^{2}x)dx$

$=\int ({\sec }^{2}x-{\sec }^{2}x{\cot }^{2}x)dx$

$=\int ({\sec }^{2}x-{\csc }^{2}x)dx$

$=\int {\sec }^{2}xdx-\int {\csc }^{2}xdx$

$=\tan x-(-\cot x)+C$

$=\tan x+\cot x+C$

$=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}+C$

$=\frac{{\sin }^{2}x+{\cos }^{2}x}{\sin x\cos x}+C$

$=\frac{2}{2}\times \frac{1}{\sin x\cos x}+C$

$=\frac{2}{\sin 2x}+C$

$=2cosec2x+C$

Thus $\int {\sec }^{2}x(1-{\cot }^{2}x)dx=2cosec2x+C$.