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Today's calculation of Integral 133

Source: Osaka University of education entrance exam 1968

August 10, 2006
calculusintegrationalgebrapolynomiallimitfunctionreal analysis

Problem Statement

Let f(x)f(x) be the polynomial with respect to x,x, and gn(x)=n−2n2∣x−12∣+∣n−2n2∣x−12∣∣.g_{n}(x)=n-2n^{2}\left|x-\frac{1}{2}\right|+\left|n-2n^{2}\left|x-\frac{1}{2}\right|\right|. Find lim⁡n→∞∫01f(x)gn(x) dx.\lim_{n\to\infty}\int_{0}^{1}f(x)g_{n}(x)\ dx.
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