Primitive of Power of Hyperbolic Cosine of a x by Hyperbolic Sine of a x

Theorem

$\displaystyle \int \cosh^n a x \sinh a x \rd x = \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$

for $n \ne -1$.

Proof

 $\displaystyle u$ $=$ $\displaystyle \cosh a x$ $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d u} {\d x}$ $=$ $\displaystyle a \sinh a x$ Derivative of $\cosh a x$ $\displaystyle \leadsto \ \$ $\displaystyle \int \cosh^n a x \sinh a x \rd x$ $=$ $\displaystyle \int \frac {u^n} a \rd u$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac 1 a \int u^n \rd u$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac 1 a \frac {u^{n + 1} } {n + 1} + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {\cosh^{n + 1} a x} {\paren {n + 1} a} + C$ substituting for $u$

$\blacksquare$