由乘积和极限求某些对数积分的一种捷径
A shortcut for evaluating some log integrals from products and limits
摘要：
在这篇短文中，我介绍了一个初等方法，用于精确计算定积分$\, \int_0^π{\ln{(\sinθ)}\,dθ}$，$\int_0^{π/2}{\ln{(\sinθ)}\,dθ}$，$\int_0^{π/2}{\ln{(\cosθ)}\,dθ}$和$\int_0^{π/2}{\ln{(\tanθ)}\,dθ} \,$的有限项。该方法包括在操纵的总和从某些产品的三角函数在有理倍数$π$的对数，把它们的形式黎曼和。由于这种方法不涉及对原语的任何搜索，因此它代表了更复杂的集成技术的一个很好的替代方案。作为奖励，我将展示如何应用该方法轻松评估$\,\int_0^1{\ln{Γ(x)} \, d x}$。
英文摘要：
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 $\,\int_0^1{\ln{Γ(x)} \, d x}$.
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