TITLE: Calculating the value of integral of $f''(x)$ when given values of $f(x)$ and $f'(x)$.
QUESTION [1 upvotes]: Here is the original question.
Suppose that $f(1)=2$, $f(4)=7$, $f'(1)=5$, $f'(4)=3$, and $f''$ is continuous.
Find the value of $\int_1^4 {xf''(x)\,dx}$.

REPLY [3 votes]: Integrate by parts 
$$\int xf''(x)dx=x\int f''(x)dx-\int\left(\frac{dx}{dx}\int f''(x)dx\right)dx$$
As $\displaystyle\frac{d(f^n(x))}{dx}=f^{n+1}(x),\int f^{n+1}(x)\ dx=f^n(x)+ K$
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