Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous.
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2026-04-06 11:39:08.1775475548
Find the value of $\int_1^4 xf''(x)dx$ when $f(1) = 2$, $f(4) = 7$, $f'(1) = 5$, $f'(4) = 3$, and $f''$ is continuous
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One may just integrate by parts: $$ \int_1^4 xf''(x)dx=\left.xf'(x)\right|_1^4-\int_1^4 f'(x)dx=4\times f'(4)-\left.f(x)\right|_1^4. $$