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I was given the following integration rule:

$$\int_{x_0}^{x_1} y(x)\ dx = h(ay_0 + by_1 + cy_2) +R $$

Where $x_i = x_0 + ih$ and $y(x_i) = y_i$

I found that $a=5/12, b=2/3$ and $c=-1/12$ in order to make the rule exact for $\{1,x,x^2\}$.

Now, I was asked to show that $R = ky^{(n)}(\xi)$ where $x_0 \le \xi \le x_2$, but I'm not sure how to do it.

I'd be glad for help!

EDIT:
The final solution is: $$R = \frac{C}{3!}y^{(3)}(\xi), \ x_0 < \xi < x_1$$

Where $$C = \int_{x_0}^{x_1} x^3 dx - h(ax_0^3 +bx_1^3 + cx_2^3) = \frac{h^4}{24}$$

How to reach this solution? Is there a theorem behind that?

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    $\int_{x_0}^{\color{red}{x_2}}$ right? – 2017-02-27
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    @Winther, It's actually $x_1$ but I wonder that's a mistake.. – 2017-02-27
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    That is strange. I don't understand why one would use a value *outside* of the integration region to estimate this integral. – 2017-02-27
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    @Winther, I've added more information regarding the solution. – 2017-02-27

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