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

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

 * $\displaystyle \int e^{a x} \cos b x \rd x = \frac {e^{a x} \cos b x} a + \frac b a \int e^{a x} \sin b x \rd x$

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: