Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \sec ^{6} x \tan x d x$

Solution:

Let $\mathrm{I}=\int \sec ^{6} \mathrm{x} \tan \mathrm{x} \mathrm{dx}$

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

Substituting, $\sec x=t \Rightarrow \sec x \tan x d x=d t$

$\Rightarrow I=\int t^{5} d t$

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

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

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

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