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In this short paper, I introduce an elementary method for exactly evaluatingthe 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 methodconsists in to manipulate the sums obtained from the logarithm of certainproducts of trigonometric functions at rational multiples of $\pi$, puttingthem in the form of Riemann sums. As this method does not involve any searchfor primitives, it represents a good alternative to more involved integrationtechniques. As a bonus, I show how to apply the method for easily evaluating$\,\int_0^1{\ln{\Gamma(x)} \, d x}$.

A shortcut for evaluating some log integrals from products and limits