Primitive of x by Sine of a x

Theorem

 * $\ds \int x \sin a x \rd x = \frac {\sin a x} {a^2} - \frac {x \cos a x} a + C$

where $C$ is an arbitrary constant.

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

let:

and let:

Then:

Also see

 * Primitive of $x \cos a x$
 * Primitive of $x \tan a x$
 * Primitive of $x \cot a x$
 * Primitive of $x \sec a x$
 * Primitive of $x \csc a x$