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I have an even and continuous function $f:[-1,1]\rightarrow\mathbb{R}$ and I want to prove that: $$\int_0^{2\pi} x f(\sin x) \, \text{d}x = 2\pi \int_0^\pi f(\sin x) \, \text{d}x$$ So I am thinking of applying u-substitution, but if I let $u=\sin x$, then I don't know how to change the $x$ for a something that contains $u$. Any ideas?

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hint......try substituting $u=\pi-x$

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