Integration involving hyperbolic functions

Question

As part of a physics problem involving a particle of mass m that slid down an inclined plane of angle $\theta$ and experienced a frictional/retarding force of $f = kmv^2$, I reduced the problem to the integrals: $\int \frac{dx}{\sqrt{1-e^{-2kx}}}=\int \sqrt{\frac{gsin(\theta )}{k}}dt$. After doing a u-substitution of $u=e^{kx}$, I eventually obtained $x(t) = \frac{1}{k} ln cosh\sqrt{gksin(\theta )}t.$ How can I show the steps to express the final equation instead as $x - x_{0}$ $= (1/k)$$lncosh[\sqrt{gksin(\theta)}$$(t - t_{0})]$.

Answer 1

You've lost an integration constant (on the right, perhaps?). Remember it's often easier to use definite integrals, to nail this stuff down to the boundary conditions.