2 x ^2 x ^2 x d x is equal to - Vidyadip

Question

$\int \frac{\cos 2 x}{\sin ^2 x \cdot \cos ^2 x} d x$ is equal to

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Answer

$\text {Let, } I=\int \frac{\cos 2 x}{\sin ^2 x \cdot \cos ^2 x} d x$
$=\int\left(\frac{\cos ^2 x-\sin ^2 x}{\sin ^2 x \cdot \cos ^2 x}\right) d x \quad\left[\because \cos 2 \theta=\cos ^2 \theta-\sin ^2 \theta\right]$
$=\int \frac{1}{\sin ^2 x} d x-\int \frac{1}{\cos ^2 x} d x=\int \operatorname{cosec}^2 x d x-\int \sec ^2 x d x$
$=-\cot x-\tan x+C$

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