quadratic drag equation by partial fractions

[matpo39]

hi, i am trying to show that

dv/(1- (v^2/v_ter^2)) = g*dt which after integrating is

v=v_ter*tanh(g*t/v_ter) (motion with quadratic drag) can also be obtained by using natural logs.

so far i have this:

letting u = v/v_ter

i can use partial fractions to get

du/(1-u^2) = 1/2 *(1/(1+u) + 1/(1-u)) *du

then using my limits of integration as 0 to u , i get

1/2* [ln(1+u) + ln(1-u)] = g*dt

then integrating the other side i get as my final equation

1/2 *[ ln(1+v/v_ter) + ln(1-v/v_ter)] = g*t

but when i tried to plug numbers into each equation the numbers didnt match.

does anyone know what i may have done wrong?
thanks

[Dr Transport]

What is \int \frac{1}{1- \frac{v^{2}}{v_{0}}} dv , look it up in an integral table, you shouldn't have to resort to partial fractions, I suspect that it will be arctanh().........