# Primitive of Hyperbolic Sine of p x by Hyperbolic Cosine of q x

## Theorem

$\displaystyle \int \sinh p x \cosh q x \ \mathrm d x = \frac {\cosh \left({p + q}\right) x} {2 \left({p + q}\right)} + \frac {\cosh \left({p - q}\right) x} {2 \left({p - q}\right)} + C$

## Proof

 $\displaystyle \int \sinh p x \cosh q x \ \mathrm d x$ $=$ $\displaystyle \int \left({\frac {\sinh \left({p x + q x}\right) + \sinh \left({p x - q x}\right)} 2}\right) \ \mathrm d x$ Simpson's Formula for Hyperbolic Sine by Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \frac 1 2 \int \sinh \left({p + q}\right) x \ \mathrm d x + \frac 1 2 \int \sinh \left({p - q}\right) x \ \mathrm d x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac 1 2 \frac {\cosh \left({p + q}\right) x} {p + q} + \frac 1 2 \frac {\cosh \left({p - q}\right) x} {p - q} + C$ Primitive of $\sinh a x$ $\displaystyle$ $=$ $\displaystyle \frac {\cosh \left({p + q}\right) x} {2 \left({p + q}\right)} + \frac {\cosh \left({p - q}\right) x} {2 \left({p - q}\right)} + C$ simplifying

$\blacksquare$