Primitive of Hyperbolic Sine of a x by Hyperbolic Cosine of a x/Proof 4

Theorem

 * $\displaystyle \int \sinh a x \cosh a x \ \mathrm d x = \frac {\sinh^2 a x} {2 a} + C$

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: