Primitive of Power of x by Hyperbolic Sine of a x

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Theorem

$\displaystyle \int x^m \sinh a x \ \mathrm d x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \ \mathrm d x + C$


Proof

With a view to expressing the primitive in the form:

$\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

\(\displaystyle u\) \(=\) \(\displaystyle x^m\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d u} {\mathrm d x}\) \(=\) \(\displaystyle m x^{m - 1}\) Derivative of Power


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle \sinh a x\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {\cosh a x} a\) Primitive of $\sinh a x$


Then:

\(\displaystyle \int x^m \sinh a x \ \mathrm d x\) \(=\) \(\displaystyle x^m \left({\frac {\cosh a x} a}\right) - \int \left({\frac {\cosh a x} a}\right) \left({m x^{m - 1} }\right) \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \ \mathrm d x + C\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


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