一元函数积分求法的技巧性总结报告.doc

摘 要 本文论述了求解一元函数积分的若干方法,使我们在解决积分问题时,可以根据具体情况,采用适当、有效的解题方法,解决许多计算繁琐的积分问题.解决好定积分与不定积分计算问题,是为更好地解决反常积分、含参变量积分、重积分以及曲线曲面积分等积分推广奠定坚实的基础.积分发展的动力来自于实际应用中的需求,它的思想萌芽可追溯到古希腊时期用以求面积和体积的穷竭法,此后,莱布尼兹通过无穷多个无穷小之和用以计算积分,直至今日,对于积分求法的技巧仍在不断的探究之中.本文在前人刻苦钻研、精益求精的基础上,从一般方法到特殊解法,再由技巧指导到具体例题的应用,由浅入深,逐渐剖析积分求解的技巧性.与此同时,在将积分与其他学科间知识,加以整合并得出新意解法的过程中,不仅锻炼了我们整合、归纳知识的能力,同时也促进了我们对学科间知识的灵活运用,使我们在自主学习、逻辑思维等方面获益匪浅. 关键词: 一元函数积分，解题技巧，特殊方法 Unary function integration method for finding the skill Abstract: This paper discusses several methods for solving integral functions of one variable, so that when we solve the integration problem, depending on the circumstances, the use of appropriate problem-solving approach to solve many complicated computing problems. Solve the problem of indefinite integral and definite integral, integral to better address the anomaly, with one variable calculus, integral and important points, such as the problem of integral curves and surfaces of the foundation. For the integration problem solving skills, our country and it is many foreign scholars have been thoroughly explored. Based on the previous assiduously, on the basis of excellence, from the general to the particular method of solution, then directed by the application of techniques to specific examples, deep integration analysis gradually solving skill. At the same time, the integration between knowledge of other disciplines, integrate and draw innovative solution in the process, not only exercise our integration, the ability to generalize the knowledge, but also to promote flexibility in the use of our inter-disciplinary knowledge, so that we benefit in terms of learning, logical thinking and so on. Keywords: Unary function integral, Common Solution, Special Skills, Other Solution 目 录 一、引言 1 二、不定积分的若干解题技巧 1 （一）利用定义求不定积分 1 （二）利用直接积分法求不定积分 2 （三）利用换元积分法求不定积分 2 1．第一换元积分法 2 2．第二换元积分法 3 （四）利用分部积分法求不定积分 5 1．“降幂”分部积分法 5 2．“降幂”积分法—矩阵积分法 6 3．“升幂”分部积分法 7 三、定积分的若干解题技巧 8 （一）利用定义求定积分 8 （二）利用牛顿—莱布尼茨公式求定积分 9 （三）利用换元积分法求定积分 10 （四）