A shortcut for evaluating some log integrals from products and limits

Abstract

In this short paper, I introduce an elementary method for exactly evaluating the definite integrals ${\int }_0^{\pi }\ln (\sin \theta )d\theta$, ${\int }_0^{\pi /2}\ln (\sin \theta )d\theta$, ${\int }_0^{\pi /2}\ln (\cos \theta )d\theta$, and ${\int }_0^{\pi /2}\ln (\tan \theta )d\theta$ in finite terms. The method consists in to manipulate the sums obtained from the logarithm of certain products of trigonometric functions at rational multiples of $\pi$, putting them in the form of Riemann sums. As this method does not involve any search for primitives, it represents a good alternative to more involved integration techniques. As a bonus, I show how to apply the method for easily evaluating ${\int }_0^1\ln \Gamma (x)dx$.

arXiv comment6 pages, no figures. Revised form. Some small corrections. Submitted to: IJMEST (06/26/2012)