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

Theorem

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

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: