Primitive of Hyperbolic Sine of a x over Power of x

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Theorem

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


Proof

With a view to expressing the problem 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:

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


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac 1 {x^n}\)
\(\ds \) \(=\) \(\ds x^{-n}\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {x^{-n + 1} } {- n + 1}\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {-1} {\paren {n - 1} x^{n - 1} }\) simplifying


Then:

\(\ds \int \frac {\sinh a x \rd x} {x^n}\) \(=\) \(\ds \sinh a x \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } - \int \paren {\frac {-1} {\paren {n - 1} x^{n - 1} } } \paren {a \cosh a x} \rd x\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac {-\sinh a x} {\paren {n - 1} x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \rd x} {x^{n - 1} }\) simplifying

$\blacksquare$


Also see


Sources

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