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

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Theorem

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

for $n \ne -1$.


Proof

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

$\blacksquare$


Also see


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