# 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.

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