I'm doing a physics problem in which I need to find the energy dissipated from air resistance. The straightforward equation which could be solved is $$\Delta E = k\int_0^T \left(\frac{\text{d}x}{\text{d}t}\right)^3\text{d}t$$ And this comes from solving $$\ddot{x} = -G\frac{M}{x^2} + k\dot{x}^2 = F_{\text{gravity}} + F_{\text{friction}}$$ Where $$\Delta E = \frac{1}{2}\left(\dot{x}(T)^2 - \dot{x}(0)^2\right) - GM\left(\frac{1}{x(T)} - \frac{1}{x(0)}\right)$$
I know I can simplify the integral slightly to
$$\Delta E = \int_{x(0)}^{x(T)} \left(\frac{\text{d}x}{\text{d}t}\right)^2\text{d}x$$ But I feel like this just changes the problem, doesn't help it.
I am also told that the frictional component provides a small perturbation to the gravitational component. I have no idea how this could help, but perhaps someone else does.