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Today's calculation of Integral 824

Source: 1999 Nara Medical University entrance exam

June 13, 2012
calculusintegrationgeometrylogarithmslimitcalculus computations

Problem Statement

In the xyxy-plane, for a>1a>1 denote by S(a)S(a) the area of the figure bounded by the curve y=(a−x)ln⁡xy=(a-x)\ln x and the xx-axis.
Find the value of integer nn for which lim⁡a→∞S(a)anln⁡a\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a} is non-zero real number.
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