How can I calculate the following integral??
$$\int \frac{1}{t}u'(t) dt$$
I thought that I could it as followed:
$$\int \frac{1}{t}u'(t) dt=\frac{1}{t}u(t)+\int \frac{1}{t^2}u(t) dt$$
but I don't know how to continue...
Or is there an other way to calculate it??
This integral has no closed form since we don't know what this function $u'(t)$ can be. For example, take $u(t)=-\cos t$, then $u'(t)=\sin t$, and thus we will get: $$\int\dfrac{\sin t}{t}\mathrm dt\,,$$ which is known to not have an antiderivative expressible in terms of elementary functions. So if a closed form for your integral existed then that would mean that a closed form for this since integral exists, which isn't possible.