Question

If $I_{n}=\int \sin^{n} x \,dx ,$ then $nI_{n}-\left(n-1\right)I_{n-2}$ equals

• A. $\sin ^{n-1} x \cos x$
• B. $\cos ^{n-1} x \sin x$
• C. $-\sin ^{n-1} x \cos x$
• D. $-\cos ^{n-1} x \sin x$

Answer 1

Answer: (C)

We know that, if
$I_{n}=\int \sin ^{n} x \,d x, $ then
$I_{n}=-\frac{\sin ^{n-1} x \cos x}{n}+\frac{n-1}{n} I_{n-2}$
where $n$ is a positive integer.
$\Rightarrow n I_{n}-(n-1) I_{n-2}=-\sin ^{n-1} x \cos x$