Thm ¶

$\displaystyle \sinh^{-1}x=\ln(x+\sqrt{x^2+1})$

$\displaystyle x\in\mathbb{R}$

Pf ¶

$\displaystyle y=\sinh^{-1}x$
i.e.
$\displaystyle \sinh y=x$
$\displaystyle \frac{e^y-e^{-y}}2=x$
$\displaystyle e^y-2x-e^{-y}=0$

$\displaystyle t=e^y$ 치환

$\displaystyle t-2x-\frac1t=0$
$\displaystyle t^2-2xt-1=0$
$\displaystyle t=e^y=x\pm\sqrt{x^2+1}$

$\displaystyle t=e^y>0$ 이므로

$\displaystyle e^y=x+\sqrt{x^2+1}$