Question

Solve:
$\int \frac{{\sec }^{8}x}{\cos ecx}dx$

Answer 1

$\int \frac{{\sec }^{8}x}{\csc x}dx=\int {\sec }^{7}x.\sec x.\sin xdx$

$=\int {\sec }^{7}x.\tan xdx$

$=\int {\sec }^{6}x.\sec x\tan xdx$

Let $t=\sec x$

$\Rightarrow dt=\sec x\tan xdx$

$=\int t^{6}dt=\frac{t^7}{7}+c$

$=\frac{1}{7}{\sec }^{7}x+c$

$\therefore \int \frac{{\sec }^{8}x}{\csc x}dx=\frac{1}{7}{\sec }^{7}x+c$