摘要
讨论并证明了含参变量x的定积分I(x)=∫0π1n(1-2xcosθ+x2)dθ对参x所有值(包括±1)的存在性和连续性,借助于定积分的定义、Riemann和的极限、函数的性质及Fourier级数和幂级数的应用,给出了计算积分I(x)的几种方法。
This article studies the existence and continuity of all the values of the variable x, including ±1, which exist in the definite integral, I(x) = ∫0πln(1-2xcosθ+ x2)dx. Different calculating methods of the integral I(x) are introduced in this article, based on using the definition of the integral, the limit of the Riemann sum, functional properties and the application of uniformly converges of power series in x and Fourier series.
