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 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving Inverse Hyperbolic Functions: $14.673$