Answer
$2$
Work Step by Step
Integration by parts formula: $\displaystyle \int fg' $ $\displaystyle dx = fg-\int f'g$ $dx$
$\displaystyle \int_1^4xf''(x)dx$
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Let $\displaystyle f = x$ and $\displaystyle g' = f''(x)$
thus $\displaystyle f' = 1$ and $\displaystyle g = f'(x)$
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$\displaystyle \int_1^4xf''(x)dx = [xf'(x)]_1^4-\int_1^4(1)f'(x)dx$
$\displaystyle [4f'(4) - (1)f'(1)] - [f(4) - f(1)]$
$\displaystyle [4(3) - 5] - (7-2)$
$\displaystyle (12-5)-5$
$\displaystyle 7-5$
$\displaystyle 2$
