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Today's calculation of Integral 184

Source: Tokyo University entrance exam/Science, Problem 6, 2007

February 25, 2007
calculusintegrationlogarithmsfunctioninequalitiesreal analysiscalculus computations

Problem Statement

(1) For real numbers x, ax,\ a such that 0<x<a,0<x<a, prove the following inequality. 2xa<∫a−xa+x1t dt<x(1a+x+1a−x).\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). (2) Use the result of (1)(1) to prove that 0.68<ln⁡2<0.71.0.68<\ln 2<0.71.
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