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The three-point quadrature rule with error term is given by $$\int_{-1}^1f(x)dx=\frac59f\left(\frac{-\sqrt{15}}5\right)+\frac89f(0)+\frac59f\left(\frac{\sqrt{15}}5\right)+kf^{(6)}(c).$$ Find $k$.

After using Lagrange's interpolation (interpolating $f$ at $\frac{-\sqrt{15}}5,0,\frac{\sqrt{15}}5$), I found that the error term should be of the form $$\int_{-1}^1\frac{f'''(c_x)}6x(x^2-\frac53)dx.$$ However, I can't use the mean value theorem, because $x(x^2-\frac53)$ changes sign in $[-1,1]$. So how to continue from here?

I think this is somewhat related to the Simpson's error terms. But the textbook I'm using (Sauer) omitted the proof.

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