Reduction Formula for Integral of Power of Sine/Corollary

Corollary to Reduction Formula for Integral of Power of Sine

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 $\ds \int \sin^n a x \rd x$.

$\ds \int \sin^n a x \rd x$

$=$

$\ds \frac 1 a \int \sin^n a x \map \rd {a x}$

$\ds $

$=$

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

$\ds $

$=$

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

$\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

$\blacksquare$