Odd positive power of sine function

Definition

This page is about functions of the form:

$x\mapsto (\sin x)^{n}$

where $n$ is an odd positive integer, i.e., $n=2k+1$ for $k$ a nonnegative integer.

In other words, the function is the composite of an odd positive power function and the sine function.

Integration

First antiderivative: as a polynomial in cosine

We consider $n=2k+1$, $k$ a nonnegative integer:

$\int \sin ^{2k+1}x\,dx$

Rewrite $\sin ^{2k+1}x=\sin ^{2k}x\sin x=(1-\cos ^{2}x)^{k}\sin x$. We get:

$\int (1-\cos ^{2}x)^{k}(\sin x)\,dx$

Set $u=\cos x$, and we get:

$\int -(1-u^{2})^{k}\,du$

This is a polynomial integration in $u$. After obtaining the answer, we plug back $u=\cos x$.

Here is the general integration in terms of binomial coefficients: [SHOW MORE]

$\int -\sum _{r=0}^{k}(-1)^{r}{\binom {k}{r}}u^{2r}\,du$

Integrating term-wise, we get:

$\sum _{r=0}^{k}\left[{\frac {(-1)^{r+1}}{2r+1}}{\binom {k}{r}}u^{2r+1}\right]+C$

Plugging back $u=\cos x$, we get:

$\sum _{r=0}^{k}\left[{\frac {(-1)^{r+1}}{2r+1}}{\binom {k}{r}}\cos ^{2r+1}x\right]+C$

Note that in all instances, the answer is an odd polynomial of the cosine function.

We consider the integration in some small cases:

$k$	$n=2k+1$	$\sin ^{n}$	Antiderivative as a polynomial in $\cos$
0	1	sine function	$\!-\cos x+C$
1	3	sine-cubed function	${\frac {\cos ^{3}x}{3}}-\cos x+C$
2	5	fifth power of sine function	$-{\frac {\cos ^{5}x}{5}}+{\frac {2\cos ^{3}x}{3}}-\cos x+C$
3	7	seventh power of sine function	${\frac {\cos ^{7}x}{7}}-{\frac {3\cos ^{5}x}{5}}+\cos ^{3}x-\cos x+C$