Primitive of x cubed by Sine of a x

Theorem

 * $\displaystyle \int x^3 \sin a x \ \mathrm d x = \left({\frac {3 x^2} {a^2} - \frac 6 {a^4} }\right) \sin a x + \left({\frac {6 x} {a^3} - \frac {x^3} a}\right) \cos a x + C$

where $C$ is an arbitrary constant.

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:

Also see

 * Primitive of $x^3 \cos a x$