If-f-0-1-f-2-3-f-2-5-find-the-value-of-0-1-x-f-2x-dx-

Question Number 154359 by ZiYangLee last updated on 17/Sep/21

If f (0) = 1, f (2) = 3, f^{'} (2) = 5,

find the value of \int_{0}^{1} x f^{″} (2 x) d x .

Answered by aleks041103 last updated on 17/Sep/21

\int {x f}^{″} (2 x) d x = \int x d (\frac{1}{2} f^{'} (2 x)) =

= \frac{1}{2} {x f}^{'} (2 x) - \frac{1}{2} \int f^{'} (2 x) d x =

= \frac{1}{2} {x f}^{'} (2 x) - \frac{1}{4} f (2 x) + C

\Rightarrow \int_{0}^{1} {x f}^{″} (2 x) d x = (\frac{1}{2} {x f}^{'} (2 x) - \frac{1}{4} f (2 x)) ∣_{0}^{1} =

= \frac{1}{2} f^{'} (2) - \frac{1}{4} (f (2) - f (0)) =

= \frac{5}{2} - \frac{2}{4} = 2