Evaluate a definite integral of log (e^{cos x})

by RoRi

December 17, 2015

Evaluate the integral

$\int_0^{\frac{\pi}{2}} \log (e^{\cos x}) \, dx.$

Since $\log (e^{\cos x}) = \cos x$ we have

$\begin{align*} \int_0^{\frac{\pi}{2}} \log (e^{\cos x}) \, dx &= \int_0^{\frac{\pi}{2}} \cos x \, dx \\ &= \sin x \Bigr \rvert_0^{\frac{\pi}{2}} \\ &= 1. \end{align*}$

Stumbling Robot

Evaluate a definite integral of log (e^{cos x})

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