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

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Theorem

$\ds \int x^m \paren {-\cosh^{-1} \frac x a} \rd x = \dfrac {x^{m + 1} } {m + 1} \paren {-\cosh^{-1} \frac x a} + \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C$

where $-\cosh^{-1}$ denotes the negative branch of the real inverse hyperbolic cosine multifunction.


Proof

\(\ds -\cosh^{-1} \frac x a\) \(=\) \(\ds -\arcosh \frac x a\) Definition of Real Inverse Hyperbolic Cosine
\(\ds \leadsto \ \ \) \(\ds \int x^m \paren {-\cosh^{-1} \frac x a} \rd x\) \(=\) \(\ds -\int x^m \arcosh \frac x a \rd x\)
\(\ds \) \(=\) \(\ds -\paren {\dfrac {x^{m + 1} } {m + 1} \arcosh \dfrac x a - \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C}\) Primitive of $x^m \arcosh \dfrac x a$
\(\ds \) \(=\) \(\ds \dfrac {x^{m + 1} } {m + 1} \paren {-\cosh^{-1} \frac x a} + \dfrac 1 {m + 1} \int \dfrac {x^{m + 1} } {\sqrt {x^2 - a^2} } \rd x + C\) Definition of Real Inverse Hyperbolic Cosine

$\blacksquare$


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