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Consider the integral $I_1=\displaystyle \int_0^\tau \frac{dx(t)}{dt} dt$ that contains the derivative $\dot{x}$.

Because $\dot{x}$ is not inside a non linear expression, one calculates that it equals $\displaystyle \int_{x(0)}^{x(\tau)} dx(t) = x(\tau)-x(0)$.

But how can one calculate an integral such as $I_2=\displaystyle \int_0^\tau \sqrt{\frac{dx(t)}{dt}} dt$? I tried partial integration, but this does not work.

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There is no useful general formula. It depends on the function $x$ and can depend on the number $\tau$ as to how nice an answer can be found.

For example, if $x(t)=t-\dfrac{t^3}3$ (so $x'(t)=1-t^2$) and $\tau=1$, then $\int_0^\tau\sqrt{x'(t)}\,\mathrm dt=\dfrac{\pi}4$ since it's the area of one quarter of a unit circle. And a formula isn't too bad for other values of $\tau$ in $[-1,1]$.

But if $x(t)=(t+1)\ln(t+1)-t$ then $x'(t)=\ln(t+1)$ and even $\tau=1$ gives you an integral that you can't write down with elementary functions (maybe not even common non-elementary functions like $\Gamma$ (see Gamma function).

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