How to calculate the integral of $x\frac{\text{d}f}{\text{d}x}$?

Question

How can we calculate this integral $ \int x \frac{\text{d} f}{\text{d} x} \,\text{d}x $ ?

I have tried both integration per partes and change of variables, but it doesn't seem to work. Of course, we don't know the $ f $ function here.

Could someone help me? Thanks in advance.

Answer 1

The most you can say is that $$\int_a^xxf'(x)\,\mathrm{d}x=xf(x)-af(a)-\int_a^xf(x)\,\mathrm{d}x.$$ Without further information about $f,$ nothing can be said of the integral on the right-hand side.

Answer 2

By parts,

$$x\,f(x)-\int f(x)\,dx.$$

That's all you can say.