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., $n = 2 k + 1$ for $k$ a nonnegative integer.

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

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

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

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

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

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

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

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]

$\int - \sum_{r = 0}^{k} (- 1)^{r} (\binom{k}{r}) u^{2 r} d u$

Integrating term-wise, we get:

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

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

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

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

We consider the integration in some small cases:

$k$	$n = 2 k + 1$	$\sin^{n}$	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	$\frac{\cos^{7} x}{7} - \frac{3 \cos^{5} x}{5} + \cos^{3} x - \cos x + C$