# Odd positive power of sine function

From Calculus

## Definition

This page is about functions of the form:

where is an odd positive integer, i.e., for 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 , a nonnegative integer:

Rewrite . We get:

Set , and we get:

This is a polynomial integration in . After obtaining the answer, we plug back .

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:

Antiderivative as a polynomial in | |||
---|---|---|---|

0 | 1 | sine function | |

1 | 3 | sine-cubed function | |

2 | 5 | fifth power of sine function | |

3 | 7 | seventh power of sine function |