Let
Integrate by parts, taking
<em>u</em> = <em>x</em> ==> d<em>u</em> = d<em>x</em>
d<em>v</em> = sin<em>ᵐ </em>(<em>x</em>) d<em>x</em> ==> <em>v</em> = ∫ sin<em>ᵐ </em>(<em>x</em>) d<em>x</em>
so that
There is a well-known power reduction formula for this integral. If you want to derive it for yourself, consider the cases where <em>m</em> is even or where <em>m</em> is odd.
If <em>m</em> is even, then <em>m</em> = 2<em>k</em> for some integer <em>k</em>, and we have
Expand the binomial, then use the half-angle identity
as needed. The resulting integral can get messy for large <em>m</em> (or <em>k</em>).
If <em>m</em> is odd, then <em>m</em> = 2<em>k</em> + 1 for some integer <em>k</em>, and so
and then substitute <em>u</em> = cos(<em>x</em>) and d<em>u</em> = -sin(<em>x</em>) d<em>x</em>, so that
Expand the binomial, and so on.