# Reduction Formula for Integral of Power of Sine/Corollary

## Corollary to Reduction Formula for Integral of Power of Sine

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

Let $a \in \R_{\ne 0}$ be a non-zero real number

Then:

$\ds \int \sin^n a x \rd x = \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x - \dfrac {\sin^{n - 1} a x \cos a x} {a n}$

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

## Proof

 $\ds \int \sin^n a x \rd x$ $=$ $\ds \frac 1 a \int \sin^n a x \map \rd {a x}$ Primitive of Function of Constant Multiple $\ds$ $=$ $\ds \frac 1 a \paren {\dfrac {n - 1} n \int \sin^{n - 2} a x \map \rd {a x} - \dfrac {\sin^{n - 1} a x \cos a x} n}$ Reduction Formula for Integral of Power of Sine $\ds$ $=$ $\ds \frac 1 a \paren {a \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x} - \dfrac {\sin^{n - 1} a x \cos a x} {a n}$ Primitive of Function of Constant Multiple $\ds$ $=$ $\ds \dfrac {n - 1} n \int \sin^{n - 2} a x \rd x - \dfrac {\sin^{n - 1} a x \cos a x} {a n}$ simplification

$\blacksquare$