A shortcut for evaluating some log integrals from products and limits
F. M. S. Lima · arXiv (Cornell University) · 2009
In this short paper, I introduce an elementary method for exactly evaluating the definite integrals $\, \int_0^π{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\sinθ)}\,dθ}$, $\int_0^{π/2}{\ln{(\cosθ)}\,dθ}$, and $\int_0^{π/2}{\ln{(\tanθ)}\,dθ} \,$ 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 $π$, 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 $\,\int_0^1{\ln{Γ(x)} \, d x}$.