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:

$\int {\sin }^{2k+1}xdx$

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]

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

Integrating term-wise, we get:

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

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

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

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	$\frac{{\cos }^{7}x}{7}-\frac{3{\cos }^{5}x}{5}+{\cos }^{3}x-\cos x+C$