Primitive of Power of x by Hyperbolic Sine of a x

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Theorem

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


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds x^m\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds m x^{m - 1}\) Derivative of Power


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \sinh a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\cosh a x} a\) Primitive of $\sinh a x$


Then:

\(\ds \int x^m \sinh a x \rd x\) \(=\) \(\ds x^m \paren {\frac {\cosh a x} a} - \int \paren {\frac {\cosh a x} a} \paren {m x^{m - 1} } \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x + C\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see


Sources