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

From ProofWiki
Jump to navigation Jump to search

Theorem

$\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x = \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4 + C$

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
\(\ds \leadsto \ \ \) \(\ds \int x \paren {-\cosh^{-1} \frac x a} \rd x\) \(=\) \(\ds -\int x \arcosh \frac x a \rd x\)
\(\ds \) \(=\) \(\ds -\paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \arcosh \dfrac x a - \dfrac {x \sqrt {x^2 - a^2} } 4 + C\) Primitive of $x \arcosh \dfrac x a$
\(\ds \) \(=\) \(\ds \paren {\dfrac {x^2} 2 - \dfrac {a^2} 4} \paren {-\cosh^{-1} \frac x a} + \dfrac {x \sqrt {x^2 - a^2} } 4\) Definition of Real Inverse Hyperbolic Cosine

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_x_by_Inverse_Hyperbolic_Cosine_of_x_over_a/Corollary&oldid=505063"