Question

Suppose that f(1=9, f(4=?3, f?(1=9, f?(4=?5, and f?? is continuous. find the value of ?41xf??(xdx.

Answer 1

Some characters in your question are not displayed properly, so I'm guessing at what it says.

"Suppose that $f(1)=9$, $f(4)=-3$, $f'(1)=9$, $f'(4)=-5$, and $f''$ is continuous. Find the value of $\displaystyle\int_1^4xf''(x)\,\mathrm dx$."

(Correct me if I'm wrong!)

Integrate by parts, letting $u=x$ and $\mathrm dv=f''(x)\,\mathrm dx$, so that $\mathrm du=\mathrm dx$ and $v=f'(x)$. Then you have

$\displaystyle\int_1^4xf''(x)\,\mathrm dx=xf'(x)\bigg|_{x=1}^{x=4}-\int_1^4f'(x)\,\mathrm dx$

Integrating once more gives

$\displaystyle\int_1^4xf''(x)\,\mathrm dx=xf'(x)\bigg|_{x=1}^{x=4}-f(x)\bigg|_{x=1}^{x=4}$
$=(4f'(4)-1f'(1))-(f(4)-f(1))$
$=(-20-9)-(-3-9)=-17$