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Today's calculation of Integral 644

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September 4, 2010
calculusintegrationtrigonometryinequalitieslogarithmscalculus computations

Problem Statement

For a constant pp such that ∫1pexdx=1\int_1^p e^xdx=1, prove that (∫1pexcos⁡x dx)2+(∫1pexsin⁡x dx)2>12.\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.
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