一道大学数学竞赛题的若干换元积分求解方法

Several Methods to Solve a Problem in College Mathematics Contest Based on Integration by Change of Variable

重积分的计算是过去十届大学生数学竞赛的常考题型,其求解需要较好的观察能力、逻辑推理能力与计算能力.重积分的计算有时非常困难,注重一题多解的教学能帮助学生从不同侧面看清其研究的重积分,进而积累起更多的重积分计算思路.本文给出第一届全国大学生数学竞赛非数学专业组试题填空题中题(1)的若干种换元积分求解方法,以期能在大学数学的一题多解教学方向能带来更多有益思考.

Problems concerning the calculation of multiple integralshave frequently occurred in college mathematics contest in China in the past decade,whose solutions rely on nice observation ability,reasoning ability and calculating ability.The problem concerning the calculation of multiple integrals can be very difficult,but if we put some emphasis on teaching multiple methods to solve the same problem,then we can help the students understand the problem from different perspectives,and thereby accumulate multiple integrals calculation ideas.We present several methods based on integration by change of variable for a problem included in College Mathematics Contest in China,with the intension to invoke more thoughts on the teaching of providing multiple methods for one problem.