I've revisited this problem several times today, and I'm wondering if there is a typo in the problem, and we are actually supposed to find:
[MATH]\int_0^1 xf'(x)\,dx[/MATH]
IBP then tells us, where:
[MATH]u=x\implies du=dx[/MATH]
[MATH]dv=f'(x)\,dx\implies v=f(x)[/MATH]
[MATH]\int_0^1 xf'(x)\,dx=\left[xf(x)\right]_0^1-\int_0^1 f(x)\,dx=2-3=-1[/MATH]
This is actually one of the given choices.