Primitive of Inverse Hyperbolic Cosine of x over a

From ProofWiki
Jump to navigation Jump to search

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 $\dfrac x {\sqrt {x^2 - a^2} }$



Also see


Sources