Evaluate the following functions : ^8 x cosec x d x

Question

Evaluate the following functions : $\int \frac{\sec ^8 x}{\operatorname{cosec} x} \cdot d x$

✓

Answer

$\begin{aligned}
\mathrm{I} & =\int \sec ^7 x \cdot \sec x \cdot \frac{1}{\operatorname{cosec} x} \cdot d x \\
& =\int \sec ^7 x \cdot \frac{1}{\cos x} \cdot \sin x \cdot d x \\
& =\int \sec ^7 x \cdot \tan x \cdot d x \\
& =\int \sec ^6 x \cdot \sec x \cdot \tan x \cdot d x \\
& \text { put } \sec x=t \\
& \therefore \quad \sec x \cdot \tan x \cdot d x=d t \\
& =\int t^6 \cdot d t \\
& =\frac{t^7}{7}+c \\
& =\frac{\sec ^7 x}{7}+c
\end{aligned}
$

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