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Symmetry

Let f, g : \mathbb{R} \rightarrow \mathbb{R} be continuous functions. If \mathcal{C}_f is symmetric around the line x=\frac{\alpha + \beta}{2} then prove that:

    \[\int_{\alpha}^{\beta} x g \left ( f(x) \right )\, \mathrm{d}x = \frac{\alpha+\beta}{2} \int_{\alpha}^{\beta}g \left ( f(x) \right )\, \mathrm{d}x\]

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