Primitive of Power of x by Inverse Hyperbolic Cosecant of x over a

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Theorem

$\displaystyle \int x^m \operatorname{csch}^{-1} \frac x a \rd x = \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\ \end{cases}$


Proof

With a view to expressing the primitive in the form:

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

let:

\(\displaystyle u\) \(=\) \(\displaystyle \operatorname{sech}^{-1} \frac x a\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\rd u} {\rd x}\) \(=\) \(\displaystyle \frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} }\) Derivative of $\operatorname{csch}^{-1} \dfrac x a$


and let:

\(\displaystyle \frac {\rd v} {\rd x}\) \(=\) \(\displaystyle x^m\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle \frac {x^{m + 1} } {m + 1}\) Primitive of Power


Then:

\(\displaystyle \int \frac {\operatorname{csch}^{-1} \dfrac x a \rd x} {x^2}\) \(=\) \(\displaystyle \left({\operatorname{csch}^{-1} \frac x a}\right) \left({\frac {x^{m + 1} } {m + 1} }\right) - \int \left({\frac {x^{m + 1} } {m + 1} }\right) \left({\frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } }\right) \rd x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle \begin{cases} \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a + \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x > 0 \\ \displaystyle \frac {x^{m + 1} } {m + 1} \operatorname{csch}^{-1} \frac x a - \frac 1 {m + 1} \int \frac {x^m} {\sqrt {x^2 + a^2} } \rd x + C & : x < 0 \\ \end{cases}\) Definition of Absolute Value

$\blacksquare$


Also see


Sources