Primitive of Inverse Hyperbolic Cosine of x over a over x

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Theorem

\(\ds \int \dfrac 1 x \arcosh \dfrac x a \rd x\) \(=\) \(\ds \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} + \dfrac 1 {2 \times 2^2} \paren {\dfrac a x}^2 + \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac a x}^4 + \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6^2} \paren {\dfrac a x}^6 + \dotsb + C\)

where $\arcosh$ denotes the real area hyperbolic cosine.


Corollary

\(\ds \int \dfrac 1 x \paren {-\cosh^{-1} \dfrac x a} \rd x\) \(=\) \(\ds -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} - \sum_{n \mathop \ge 1} \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C\)
\(\ds \) \(=\) \(\ds -\dfrac 1 2 \ln^2 \paren {\dfrac {2 x} a} - \dfrac 1 {2 \times 2^2} \paren {\dfrac a x}^2 - \dfrac {1 \times 3} {2 \times 4 \times 4^2} \paren {\dfrac a x}^4 - \dfrac {1 \times 3 \times 5} {2 \times 4 \times 6 \times 6^2} \paren {\dfrac a x}^6 + \dotsb + C\)

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


Proof

For $\arcosh \dfrac x a > 0$:

\(\ds \arcosh \dfrac x a\) \(=\) \(\ds \ln \frac {2 x} a - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } \paren {\frac a x}^{2 n} }\) Power Series Expansion for Real Area Hyperbolic Cosine
\(\ds \leadsto \ \ \) \(\ds \frac 1 x \arcosh \dfrac x a\) \(=\) \(\ds \dfrac 1 x \ln \frac {2 x} a - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } a^{2 n} \paren {\frac 1 x}^{2 n + 1} }\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 x \arcosh \dfrac x a \rd x\) \(=\) \(\ds \int \paren {\dfrac 1 x \ln \frac {2 x} a - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } a^{2 n} \paren {\frac 1 x}^{2 n + 1} } } \rd x\)
\(\ds \) \(=\) \(\ds \int \frac 1 x \ln \frac {2 x} a \rd x - \paren {\sum_{n \mathop = 1}^\infty \int \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } a^{2 n} \paren {\frac 1 x}^{2 n + 1} \rd x}\) Fubini's Theorem
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac {2 x} a} - \paren {\sum_{n \mathop = 1}^\infty \int \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} } a^{2 n} \paren {\frac 1 x}^{2 n + 1} \rd x} + C\) Primitive of $\dfrac {\ln x} x$: Corollary
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac {2 x} a} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 n \paren {-2 n} } a^{2 n} \paren {\dfrac 1 x}^{2 n} } + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac 1 2 \map {\ln^2} {\dfrac {2 x} a} + \sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n}^2} \paren {\frac a x}^{2 n} + C\) rearranging

$\blacksquare$


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