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limit integral with natural parameter is invariant

Source: VJIMC 2002 2.4

July 14, 2021
calculusintegration

Problem Statement

Prove that lim⁡n→∞n2(∫011+xnndx−1)=π212.\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.
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