Find $\sinh^{-1}x$

Question

The hyperbolic sine function, $\sinh(x)$ , is defined by the equation:

$$ \sinh(x) = \frac {e^x-e^{-x}} {2}$$ Find a formula for its inverse, $$ \sinh^{-1}(x) $$

Answer 1

Let $y=\sinh^{-1}(x)$. $$ x={e^y-e^{-y}\over2}\implies e^{2y}-2xe^{y}-1=0\implies e^{y}=x+\sqrt{x^2+1}\\ \implies\sinh^{-1}(x)=\ln(x+\sqrt{x^2+1}) $$