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Today's calculation of Integral 844

Source: 2012 Mie University entrance exam/Medicine

August 22, 2012
calculusintegrationlimitcalculus computations

Problem Statement

Let α\alpha be a solution satisfying the equation ∣x∣=e−x.|x|=e^{-x}. Let In=∫0α(xe−nx+αxn−1)dx (n=1, 2, ⋯ ).I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots). Find lim⁡n→∞n2In.\lim_{n\to\infty} n^2I_n.
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