Odd positive power of sine function

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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.


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^2x)^k \sin x. We get:

\int (1 - \cos^2x)^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]

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^3x}{3} - \cos x + C
2 5 fifth power of sine function  - \frac{\cos^5x}{5} + \frac{2\cos^3x}{3} - \cos x + C
3 7 seventh power of sine function \frac{\cos^7x}{7} - \frac{3 \cos^5x}{5} + \cos^3x - \cos x + C