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Today's calculation of Integral 178

Source: Meiji University entrance exam 2007

February 7, 2007
calculusintegrationfunctiontrigonometrylimitcalculus computations

Problem Statement

Let f(x)f(x) be a differentiable function such that f′(x)+f(x)=4xe−xsin⁡2x,  f(0)=0.f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0. Find lim⁡n→∞∑k=1nf(kπ).\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).
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