Primitive of Power of x by Sine of a x

Theorem

 * $\displaystyle \int x^m \sin a x \ \mathrm d x = \frac {-x^m \cos a x} a + \frac {m x^{m - 1} \sin a x} {a^2} - \frac {m \left({m - 1}\right)} {a^2} \int x^{m - 2} \sin a x \ \mathrm d x$

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

let:

and let:

Then:

Similarly, let:

and let:

Then:

So:

Also see

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