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`