I have an integral Over a particular bound.
$$\int_{V_o}^{V_f} \frac{dv}{1 - \frac{v^2}{v_t^2}}$$
Can this integral Be that of an Inverse Hyperbolic Tangent, even though it looks like it can be broken up using partial fractions? It just so happens that denominator will always be greater than zero. The reason I ask is because it comes up in my physics book pertaining to terminal velocity, and the author kind of breezes past this part.
$$\int\limits_{V_0}^{V_f}\frac{dv}{1-\frac{v^2}{v_t^2}}=\frac{1}{2}\left((-v_t)\int\limits_{V_0}^{V_f}\left[\frac{-\frac{1}{v_t}}{1-\frac{v}{v_t}}-\frac{\frac{1}{vt}}{1+\frac{v}{v_t}}\right]dv\right)=-\frac{v_t}{2}\left[\log\left(1-\frac{v}{v_t}\right)-\log\left(1+\frac{v}{v_t}\right)\right]=\frac{v_t}{2}\log\left(\frac{1+\frac{v}{v_t}}{1-\frac{v}{v_t}}\right)=\left.\frac{v_t}{2}\log\frac{v_t+v}{v_t-v}\right|_{V_0}^{V_f}=\ldots$$