Primitive of x cubed by Sine of a x

Theorem

 * $\displaystyle \int x^3 \sin a x \rd x = \paren {\frac {3 x^2} {a^2} - \frac 6 {a^4} } \sin a x + \paren {\frac {6 x} {a^3} - \frac {x^3} a} \cos a x + C$

where $C$ is an arbitrary constant.

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

let:

and let:

Then:

Also see

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