Primitive of Exponential of a x by Sine of b x/Lemma

Lemma for Primitive of $e^{a x} \sin b x$

 * $\displaystyle \int e^{a x} \sin b x \ \mathrm d x = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \ \mathrm d x$

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: