Mathematics – History and Overview
Scientific paper
2009-09-03
Mathematics
History and Overview
9 pages, no figures. Submitted to: Coll. Math. Journal (01/26/2009)
Scientific paper
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 clearly represents a good alternative to more involved integration techniques. As a bonus, I show how to apply the method for easily evaluating $\int_0^1{\ln{\Gamma(x)} d x}$.
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