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

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Theorem

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


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:

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


Then:

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

$\blacksquare$



Also see


Sources