A shortcut for evaluating some log integrals from products and limits

TL;DR

This paper presents a simple, elementary method for exactly evaluating certain definite integrals involving logarithms of sine, cosine, and tangent functions, using manipulations of sums and Riemann sums, offering an alternative to traditional techniques.

Contribution

The paper introduces a novel elementary approach to evaluate specific log-trigonometric integrals and related integrals involving the gamma function, avoiding complex primitive searches.

Findings

Exact evaluations of key log-trigonometric integrals.

Method simplifies calculations without primitive search.

Application to gamma function integral.

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 $\,\int_0^1{\ln{\Gamma(x)} \, d x}$.