Primitive of Sine of a x over Power of x

Theorem

 * $\ds \int \frac {\sin a x} {x^n} \rd x = \frac {-\sin a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cos a x} {x^{n - 1} } \rd x$

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

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\cos a x} {x^n}$