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

From ProofWiki
Jump to navigation Jump to search

Theorem

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

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


Proof

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

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Primitive_of_Power_of_x_by_Inverse_Hyperbolic_Secant_of_x_over_a/Corollary&oldid=505097"