Question

Suppose that f(1) = 2, f(4) = 5, f'(4) =3 and f'' is continuous. Find the value of

Answer 1

\[\int\limits_{1}^{4}xf''(x)dx\]

Answer 2

Integrate by parts:
\[\begin{matrix}u=x&&&dv=f''(x)\,dx\\
du=dx&&&v=f'(x)\end{matrix}\]
which gives
\[\int_1^4xf''(x)\,dx=\left[xf'(x)\right]_1^4-\int_1^4f'(x)\,dx=4f'(4)-f'(1)-(f(4)-f(1))\]
Are you given \(f'(1)\) ?

Answer 3

no f'(1) is not given

Answer 4

a peer told me the professor added that f'(1) = 0

Answer 5

okay then I would just need to plug it in.

Answer 6

Well that's convenient. I was starting to wonder if there was some theorem that would have allowed you to determine the value using what you were given.

Answer 7

Maybe there is, but thanks. How did you know to use integration by parts?

Answer 8

Mainly by the fact that you're given values of the derivative at the endpoints of the interval \([1,4]\). IBP provides a way of reducing the order of the derivative and changing the integral of the second derivative to an expression containing an integral of the first derivative.

Answer 9

The form of the given integral is also highly reminiscent of the integral \(\int xe^x\,dx\), which is usually evaluated with IBP.

Answer 10

Ah, thanks for the explanation!

Answer 11

You're welcome