Question

$\int \frac{co{s}^{n-1}}{si{n}^{n+1}}dx,n\ne 0$ is ____

• A. $\frac{co{t}^{n}x}{n}$
• B. $\frac{-co{t}^{n-1}x}{n-1}$
• C. $\frac{-co{t}^{n}x}{n}$
• D. $\frac{co{t}^{n-1}x}{n-1}$

Answer 1

The correct option is C $\frac{-co{t}^{n}x}{n}$

$\int \frac{co{s}^{n-1}x}{si{n}^{n+1}x}dx$
$=\int (\frac{cosx}{sinx}{)}^n.\frac{1}{cosx.sinx}.dx$
$=\int co{t}^{n}x.\frac{sinx}{cosx}.\frac{1}{si{n}^{2}x}dx$
$=\int co{t}^{n}x.tanx.cose{c}^{2}xdx$
$=\int co{t}^{n-1}x.cose{c}^{2}xdx$
Let $cot\left(x\right)=t$, then $cose{c}^{2}xdx=-dt$
Hence
$I=-\int t^{n-1}.dt$
$=\frac{-t^n}{n}$
$=-\frac{co{t}^{n}x}{n}$