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Today's calculation of Integral 128

Source: Miyazaki University entrance exam/medical 2006

July 26, 2006
calculusintegrationlogarithmstrigonometrycalculus computations

Problem Statement

Prove the following inequality. −π3ln⁡2+π381<∫0π3ln⁡(cos⁡x)dx<−π3162.-\frac{\pi}{3}\ln 2+\frac{\pi^{3}}{81}<\int_{0}^{\frac{\pi}{3}}\ln (\cos x) dx<-\frac{\pi^{3}}{162}.
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