Primitive of Power of Hyperbolic Cosecant of a x

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Theorem

$\ds \int \csch^n a x \rd x = \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x + C$

for $n \ne -1$.


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:

\(\ds u\) \(=\) \(\ds \csch^{n - 2} a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -a \paren {n - 2} \csch^{n - 3} a x \csch a x \coth a x\) Derivative of Power, Derivative of $\csch a x$, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds -a \paren {n - 2} \csch^{n - 2} a x \coth a x\)


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \csch^2 a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {-\coth a x} a\) Primitive of $\csch^2 a x$


Then:

\(\ds \int \csch^n a x \rd x\) \(=\) \(\ds \int \csch^{n - 2} a x \csch^2 a x \rd x\)
\(\ds \) \(=\) \(\ds \csch^{n - 2} a x \paren {\frac {-\coth a x} a}\) Integration by Parts
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \int \paren {\frac {-\coth a x} a} \paren {-a \paren {n - 2} \csch^{n - 2} a x \coth a x} \rd x\)
\(\ds \) \(=\) \(\ds \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \coth^2 a x \csch^{n - 2} a x \rd x\) simplifying
\(\ds \) \(=\) \(\ds \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \paren {1 + \csch^2 a x} \csch^{n - 2} a x \rd x\) Difference of $\coth^2$ and $\csch^2$
\(\ds \) \(=\) \(\ds \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^n a x \rd x\) Linear Combination of Primitives
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \paren {n - 2} \int \csch^{n - 2} a x \rd x\)
\(\ds \paren {n - 1} \int \csch^n a x \rd x\) \(=\) \(\ds \frac {-\csch^{n - 2} a x \coth a x} a - \paren {n - 2} \int \csch^{n - 2} a x \rd x\) gathering terms
\(\ds \int \csch^n a x \rd x\) \(=\) \(\ds \frac {-\csch^{n - 2} a x \coth a x} {a \paren {n - 1} } - \frac {n - 2} {n - 1} \int \csch^{n - 2} a x \rd x\) dividing by $n - 1$

$\blacksquare$


Also see


Sources

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