Evaluate the following integrals:


Evaluate the following integrals:

$\int \tan ^{5} x \sec ^{4} x d x$


Let $I=\int \tan ^{5} x \sec ^{4} x d x$

$\Rightarrow I=\int \tan ^{5} x \sec ^{2} x \sec ^{2} x d x$

$\Rightarrow I=\int \tan ^{5} x\left(1+\tan ^{2} x\right) \sec ^{2} x d x$

$\Rightarrow I=\int\left(\tan ^{5} x+\tan ^{7} x\right) \sec ^{2} x d x$

Let $\tan x=t$, then

$\Rightarrow \sec ^{2} x d x=d t$

$\Rightarrow I=\int\left(t^{5}+t^{7}\right) d t$

$\Rightarrow I=\frac{t^{6}}{6}+\frac{t^{8}}{8}+c$

$\Rightarrow I=\frac{\tan ^{6} x}{6}+\frac{\tan ^{8} x}{8}+c$

Therefore, $\int \tan ^{5} x \sec ^{4} x d x=\frac{\tan ^{6} x}{6}+\frac{\tan ^{8} x}{8}+c$

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