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

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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

$\blacksquare$


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