Some characters in your question are not displayed properly, so I'm guessing at what it says.
"Suppose that [tex]f(1)=9[/tex], [tex]f(4)=-3[/tex], [tex]f'(1)=9[/tex], [tex]f'(4)=-5[/tex], and [tex]f''[/tex] is continuous. Find the value of [tex]\displaystyle\int_1^4xf''(x)\,\mathrm dx[/tex]."
(Correct me if I'm wrong!)
Integrate by parts, letting [tex]u=x[/tex] and [tex]\mathrm dv=f''(x)\,\mathrm dx[/tex], so that [tex]\mathrm du=\mathrm dx[/tex] and [tex]v=f'(x)[/tex]. Then you have
[tex]\displaystyle\int_1^4xf''(x)\,\mathrm dx=xf'(x)\bigg|_{x=1}^{x=4}-\int_1^4f'(x)\,\mathrm dx[/tex]
Integrating once more gives
[tex]\displaystyle\int_1^4xf''(x)\,\mathrm dx=xf'(x)\bigg|_{x=1}^{x=4}-f(x)\bigg|_{x=1}^{x=4}[/tex]
[tex]=(4f'(4)-1f'(1))-(f(4)-f(1))[/tex]
[tex]=(-20-9)-(-3-9)=-17[/tex]
