Evaluate ^3 x+ ^3 x ^2 x ^2 x d x. - Vidyadip

Question

Evaluate $\int \frac{\sin ^3 x+\cos ^3 x}{\sin ^2 x \cos ^2 x} d x$.

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Answer

$\int \frac{\sin ^3 x+\cos ^3 x}{\sin ^2 x \cos ^2 x}\ d x$
$=\int\left(\frac{\sin ^3 x}{\sin ^2 x \cos ^2 x}+\frac{\cos ^3 x}{\sin ^2 x \cos ^2 x}\right) d x$
$=\int\left(\frac{\sin x}{\cos ^2 x}+\frac{\cos x}{\sin ^2 x}\right) d x$
$=\int(\tan x \sec x+\cot x \operatorname{cosec} x)\ d x$
$=\int \tan x \sec x\ d x+\int \cot x \operatorname{cosec} x\ d x$
$=\sec x-\operatorname{cosec} x+C$

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