Primitive of Sine of a x over Power of x

Theorem

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

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:

Also see

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