Suppose that 𝑓(1)=8, 𝑓(4)=7, 𝑓 '(1)=9, 𝑓 '(4)=−3, and 𝑓 '' is continuous. Find the value of the definite integral ∫𝑥𝑓″(𝑥) 𝑑𝑥 on the interval [1,4].

Answer 1

integrates by parts:

U = x dV = f''

dU = dx V = f'

The integral becomes:

x f' - integral f' dx

= x f' - f

limit as x->4 : 4 * f'(4) - f(4) = 4 * -3 - 7

= -12-7 = -19

limit as x->1 : 1 * f'(1) - f(1) = 1 * 9 - 8 = 1

subtracting: -19-1 = -20