Problem

$$y = \operatorname{artanh}\left(\frac{x}{\sqrt{1+x^2}}\right)$$

(a) Show that

$$\frac{dy}{dx} = \frac{1}{\sqrt{1+x^2}}$$

(b) Hence, by integrating the result in part (a), prove that

$$\operatorname{artanh}\left(\frac{x}{\sqrt{1+x^2}}\right) = \operatorname{arsinh} x$$