How to integrate by parts without having an analytic antiderivative?

Question

higher mathematics

There is a bad function F(x) for which it is impossible (or inexpedient) to obtain an antiderivative. We solve numerically. In some problem, it is required to repeatedly calculate a certain functional containing
\int_O F(x) dx
and
\int_O x * F(x) dx
There is a temptation for the second formula to use integration by parts

\int_O x * F(x) dx = x * \int_O F (x) dx - \int_O \int F(x) dx dx

What's the right way to do this trick? Or is there a better option?

Answer 1

Well, what about the derivative dF / dx, maybe it will be easier there? Maybe it would be better to take the integral by parts "in the other direction". In differentials:
xFdx = (just showing in brackets) = F(xdx) = (in parts) = x^2/2 *dF - d( Fx^2/2 ).