Question Detail
Question
Given f(0)=1, f'(0)=5, f(2)=3, f'(2)=7, and f, f' exist, find the value of ∫₀² 3xf''(x) dx
Correct Answer
Option D is the correct answer.
Detailed Explanation
Integrate by parts with u=3x, dv=f''(x)dx: ∫3xf''(x)dx = 3xf'(x) - ∫3f'(x)dx = 3xf'(x) - 3f(x). Evaluating from 0 to 2: [3(2)(7)-3(3)] - [3(0)(5)-3(1)] = (42-9)-(0-3) = 33+3 = 36.
Hint
Integrate by parts treating 3x as u and f''(x)dx as dv, then use the given function values.