# Primitive of Inverse Hyperbolic Cosine of x over a

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## Theorem

$\displaystyle \int \cosh^{-1} \frac x a \ \mathrm d x = \begin{cases} x \cosh^{-1} \dfrac x a - \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a > 0 \\ x \cosh^{-1} \dfrac x a + \sqrt {x^2 - a^2} + C & : \cosh^{-1} \dfrac x a < 0 \end{cases}$

## 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:

 $\displaystyle u$ $=$ $\displaystyle \cosh^{-1} \frac x a$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle \frac 1 {\sqrt {x^2 - a^2} }$ Derivative of $\cosh^{-1} \dfrac x a$

and let:

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

Then:

 $\displaystyle \int \cosh^{-1} \frac x a \ \mathrm d x$ $=$ $\displaystyle x \cosh^{-1} \frac x a - \int x \left({\frac 1 {\sqrt {x^2 - a^2} } }\right) \ \mathrm d x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle x \cosh^{-1} \frac x a - \int \frac {x \ \mathrm d x} {\sqrt {x^2 - a^2} } + C$ simplifying $\displaystyle$ $=$ $\displaystyle x \cosh^{-1} \frac x a - \sqrt {x^2 + a^2} + C$ Primitive of $\dfrac x {\sqrt {x^2 - a^2} }$