利用概率方法计算两类无穷积分

Calculating of Two Classes of Infinite Integrals with Probabilistic Method

提出了两类无穷积分0()dx mP x e xαγ+∞+∫和20()dx x nQ x e xαβγ+∞++∫,通过递归并借助概率统计中的伽玛分布和正态分布的有关概念和性质得到其解法,并给出了结果的一般表达式.该方法同时为具有0()dxf x e xαγ+∞+∫或20()dx xg x e xαβγ+∞++∫形式,但无法计算的积分提供了近似的计算方法.

In this paper, two classes of infinite integrals ∫＋∞0 Pm （x）eαx＋γdx and ∫＋∞0 Qn（x）eαx2＋βx＋γdx are put forward and solved. The general expressions of the results are derived through recursion in combination with the concept and properties concerning gamma distribution and normal distribution. The solutions proposed in this paper provide the approximating method for calculation of the integrals that bear the unsolvable form of ∫＋∞0f（x）eαx＋γdx or ∫＋∞0g（x） eαx2＋βx＋γdx.