Reduction Formula for Integral of Power of Cosine

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Then:
 * $\displaystyle \int \cos^n x \ \mathrm d x = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n \int \cos^{n - 2} x \ \mathrm d x$

is a reduction formula for $\displaystyle \int \cos^n x \ \mathrm d x$.

Proof
With a view to expressing the problem in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then:

Also see

 * Reduction Formula for Integral of Power of Sine