Question

Solve $\int \frac{\cos x}{{\sin }^{7}x}dx$

Answer 1

$I=\int \frac{cosx}{si{n}^{7}x}dx$ (1)

Let $\sin x=t$

Differentiate above equation with respect to $x$

$\frac{d\left(\sin x\right)}{dx}=\frac{dt}{dx}$

$\cos x=\frac{dt}{dx}$

$\cos xdx=dt$

Substitute $\cos xdx=dt$ and $\sin x=t$ in equation (1).

$I=\int \frac{dt}{t^7}$

$=\int t^{-7}dt[\int x^{n}dx=\frac{x^{n+1}}{n+1}]$

$=-\frac{t^{-6}}{6}+C$ (2)

Substitute $t=\sin x$ in equation (2).

$I=-\frac{{\sin }^{-6}x}{6}+C$

Thus, $\int \frac{cosx}{si{n}^{7}x}dx$ is $-\frac{{\sin }^{-6}x}{6}+C$ where $C$ is integration constant.