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Today's calculation of Integral 466

Source: 2009 Osaka University entrance exam

June 29, 2009
calculusintegrationlogarithmslimitinequalitiescalculus computations

Problem Statement

For n \equal{} 1,\ 2,\ 3,\ \cdots, let (pn, qn) (pn>0, qn>0) (p_n,\ q_n)\ (p_n > 0,\ q_n > 0) be the point of intersection of y \equal{} \ln (nx) and \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1. (1) Show that 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2} to find lim⁡n→∞qn \lim_{n\to\infty} q_n. (2) Find lim⁡n→∞n∫1npnln⁡(nx) dx \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx.
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