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

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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 $u$

$\blacksquare$


Also see


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