Question
A function y = f(x) is continuous and differentiable on the interval (0, 7). If ∫(5, 1) f(x) dx = 10, f(1) = 3, and f(5) = -5, find ∫(5, 1) xf'(x) dx.
Answers
oobleck
use integration by parts, with
u = x, du = dx
dv = f'(x) dx, v = f(x)
thus,
∫xf'(x) dx = xf(x) - ∫f(x) dx
so now just plug in your numbers
u = x, du = dx
dv = f'(x) dx, v = f(x)
thus,
∫xf'(x) dx = xf(x) - ∫f(x) dx
so now just plug in your numbers