Question

Let f be twice differentiable with f(0)=3 f(1)=2, and f′(1)=4. Evaluate the integral ∫[1,0] xf″(x)dx

Answer 1

use integration by parts

u = x, dv = f"(x) dx
du = dx, v = f'(x)
∫xf″(x) dx = xf'(x) - ∫f'(x) dx = xf'(x) - f(x)
Now evaluate at the limits and subtract.