# How to calculate an integral containing a nonlinear function of a derivative?

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.

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).