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Prove that the value of this integration is 0

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020
integrationcalculusperiodic function

Problem Statement

Let f:R→Rf : \mathbb{R} \to \mathbb{R} a periodic and continue function with period TT and F:R→RF : \mathbb{R} \to \mathbb{R} antiderivative of ff. Then ∫0T[F(nx)−F(x)−f(x)(n−1)T2]dx=0\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0
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