Primitive of Inverse Hyperbolic Cosecant of x over a

From ProofWiki
Jump to navigation Jump to search

Theorem

$\displaystyle \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x = \begin{cases} x \operatorname{csch}^{-1} \dfrac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\ x \operatorname{csch}^{-1} \dfrac x a - a \sinh^{-1} \dfrac x a + C & : x < 0 \end{cases}$


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:

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


and let:

\(\displaystyle \frac {\mathrm d v} {\mathrm d x}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \implies \ \ \) \(\displaystyle v\) \(=\) \(\displaystyle x\) Primitive of Constant


Then:

\(\displaystyle \int \operatorname{csch}^{-1} \frac x a \ \mathrm d x\) \(=\) \(\displaystyle x \operatorname{csch}^{-1} \frac x a - \int x \left({\frac {-a} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } }\right) \ \mathrm d x + C\) Integration by Parts
\(\displaystyle \) \(=\) \(\displaystyle x \operatorname{csch}^{-1} \frac x a + a \int \frac {x \ \mathrm d x} {\left\vert{x}\right\vert \sqrt{a^2 + x^2} } + C\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle x \operatorname{csch}^{-1} \frac x a \begin{cases} \displaystyle \mathop + a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x > 0 \\ \displaystyle \mathop - a \int \frac {\mathrm d x} {\sqrt{a^2 + x^2} } + C & : x < 0\end{cases}\) Definition of Absolute Value
\(\displaystyle \) \(=\) \(\displaystyle \begin{cases} \displaystyle x \operatorname{csch}^{-1} \frac x a + a \sinh^{-1} \dfrac x a + C & : x > 0 \\ \displaystyle x \operatorname{csch}^{-1} \frac x a - a \sinh^{-1} \dfrac x a + C & : x < 0 \end{cases}\) Primitive of $\dfrac 1 {\sqrt{a^2 + x^2} }$

$\blacksquare$


Also see


Sources