Reduction Formula for Integral of Power of Cosine

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

Let:
 * $I_n := \ds \int \cos^n x \rd x$

Then:
 * $I_n = \dfrac {\cos^{n - 1} x \sin x} n + \dfrac {n - 1} n I_{n - 2}$

is a reduction formula for $\ds \int \cos^n x \rd x$.

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

let:

and let:

Then:

Also see

 * Reduction Formula for Integral of Power of Sine
 * Reduction Formula for Integral of Power of Tangent