Question

$\int \frac{{\sec }^{8}x}{cosecx}dx$.

Answer 1

$I=\int \frac{{\sec }^{8}x}{cosecx}dx$

$I=\int \frac{\sin x}{{\cos }^{8}x}dx$

$u=\cos x\Rightarrow du=-\sin xdx$

$\Rightarrow I=\int \frac{-du}{u^8}$

$=\frac{u^{-7}}{7}+c$

$=\frac{(\cos x{)}^{-7}}{7}+c$

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