I want to find the closed form of the integral
$$\int \dot{x} dx \tag{1}$$
where $\dot{x} = \frac{dx}{dt}$. I think there should be an analytical solution. I know
$$\int \dot{x} dx = \int \dot{x}^2 dt \tag{2}$$
and maybe using the equation $\dot{x}^2 = (x + \dot{x})^2 - x^2 - 2x\dot{x}$, but then I don't know how to go further. I would appreciate if you could help me with this. I have also asked this question here on Reddit.
P.S.
I'm not entirely sure but maybe this is the solution:
If we consider that $\dot{x} = \frac{dx}{dt}$ then
$$\int \dot{x} dx = \frac{ d}{dt} \left(\int x dx \right) \tag{3}$$
which yields:
$$\int \dot{x} dx = x \dot{x} \tag{4}$$