how calculate integral $\int \dot{x} \; dx$ Let $x$ depend on $t$. $\dot{x}$ is derivative $x$ over $t$. I want to calculate the integral $\int \dot{x} \; dx$. I asked similar question about differentiation here. Any thoughts and ideas are appreciated. Thank you!
The integral $\int \dot x dx$ cannot be evaluated explicitely unless the form of the function $x(t)$ is also given. This can be easily understood in the following way

$$\int \dot x dx=\int (\dot x)^2 dt$$

that cannot be furtherly explicited. This kind of computations generally come out from studies on mechanics with dissipative systems. If you have a differential equation like

$$\ddot x=-\dot x+F(x)$$

you can multiply both sides by $\dot x$ and integrating obtain

$$\int dt \frac{d}{dt}\left(\frac{{\dot x}^2}{2}\right)-\int dx F(x)=-\int \dot x dx$$

that cannot be reduced anymore even if lhs can be expressed through an energy integral.
You can write it as $\int \frac{dx}{dt}dx$ which, assuming appropriate smoothness conditions on $\dot{x}$ is the same as $\frac{d}{dt}\int x dx = \frac{d}{dt} (\frac{x^2}{2} + C) = x\dot{x}$