Reduction Formula for Integral of Power of Sine

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

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

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

Proof
Let $n \ge 2$.

Let:
 * $\displaystyle I_n := \int \sin^n x \ \mathrm d x$

Then:

thus demonstrating the identity for all $n \ge 2$.

When $n = 1$ this degenerates to:

From Integral of Sine Function this shows that the identity still holds for $n = 1$.