一类三角函数有理式定积分的无穷级数算法

Series Algorithm for a Kind of Definite Integrals with Trigonometric Rational Functions

本文研究了一类被积函数为三角函数有理式的定积分的计算问题,其有理式的分子是正弦函数与余弦函数的高次方,分母为正弦函数或余弦函数的一次函数.利用泰勒级数将被积分函数展开成正弦函数与余弦函数的无穷级数,然后逐项积分得到一个数项级数.对数项该级数增补一些项后,将其还原为某个已知函数的泰勒展开式在某一点的值.给出的几个算例显示该方法对这类三角函数有理式定积分的计算十分有效和便捷.

This paper investigates a computing problem of a kind of definite integrals with trigonometric rational functions, whose numerator is higher power of sine functions and cosine functions, and whose denominator is a linear function of sine or cosine functions. The integrands are expanded to series of sine and cosine functions by using Taylor＇ expansion, and then a number series is obtained after integrating term by term. As we add some terms into this number series, it can be restored as a function value at a certain point. The given examples show that the proposed method is very effective and efficient for computation of integrals with considered trigonometric rational function.