Odd positive power of sine function
This page is about functions of the form:
where is an odd positive integer, i.e., for a nonnegative integer.
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|
|2||5||fifth power of sine function|
|3||7||seventh power of sine function|