Primitive of Power of x by Sine of a x

Theorem

 * $\displaystyle \int x^m \sin a x \rd x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \paren {m - 1} } {a^2} \int x^{m - 2} \sin a x \rd x$

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

let:

and let:

Then:

Similarly, let:

and let:

Then:

So:

Also see

 * Primitive of $x^m \cos a x$