Evaluate: ^3 x ^3 x d x - Vidyadip

Question

Evaluate: $\int \tan ^3 x \sec ^3 x d x$

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Answer

Let $I=\int \tan ^3 x \sec ^3 x d x$, then we have
$I=\int \tan ^2 x \sec ^2 x(\sec x \tan x) d x=\int\left(\sec ^2 x-1\right) \sec ^2 x(\sec x \tan x) d x$
Substituting sec x = t and sec x tan x dx = dt, we obtain
$I =\int\left( t ^2-1\right) t ^2 dt =\int\left( t ^4- t ^2\right) dt =\frac{t^5}{5}-\frac{t^3}{3}+C=\frac{1}{5} \sec ^5 x-\frac{1}{3} \sec ^3 x+C$

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