Hello. I've been using all the time primitivization by parts in the following way:

P (f g) = (P f)g - P ((P g) g')

Which I thought that would correctly translate to integration by parts like the following:

It seems that in the end of the expression, instead of integration f(x) from a to b, I should just let it be f(x), as it is already inside an integral. But I don't get why. Could anyone help? Thanks