Derivative of Real Area Hyperbolic Sine of x over a/Corollary 2

Theorem

$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$

$\ds -\map \arsinh {\frac x a}$

$=$

$\ds \map \arsinh {-\frac x a}$

$\ds $

$=$

$\ds \map \ln {-\paren {\frac x a} + \sqrt {\paren {-\frac x a}^2 + a^2} }$

$\ds $

$=$

$\ds \map \ln {-\frac x a + \dfrac 1 a \sqrt {x^2 + a^2} }$

$\ds $

$=$

$\ds \map \ln {-\dfrac 1 a \paren {x - \sqrt {x^2 + a^2} } }$

$\ds $

$=$

$\ds \map \ln {-x + \sqrt {x^2 + a^2} } - \ln a$

$\ds $

$=$

$\ds \ln \size {x - \sqrt {x^2 + a^2} } - \ln a$

as $\sqrt {x^2 + a^2} > -x$

$\ds \leadsto \ \ $

$\ds \map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } }$

$=$

$\ds \map {\dfrac \d {\d x} } {-\map \arsinh {\frac x a} + \ln a}$

$\ds $

$=$

$\ds -\dfrac 1 {\sqrt {x^2 + a^2} } + \map {\dfrac \d {\d x} } {\ln a}$

$\ds $

$=$

$\ds -\dfrac 1 {\sqrt {x^2 + a^2} } + 0$

$\blacksquare$

1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $14$.