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Today's calculation of Integral 792

Source: 2012 Kyoto University entrance exam/Science, problem 1

February 26, 2012
calculusintegrationlimitlogarithmstrigonometrycalculus computations

Problem Statement

Answer the following questions:
(1) Let aa be positive real number. Find lim⁡n→∞(1+an)1n.\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.
(2) Evaluate ∫131x2ln⁡1+x2dx.\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.
35 points
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