不定积分的求解方法论文

重庆三峡学院毕业设计（论文）题目：归结不定积分的求解方法

专业：数学与应用数学

年级：2010级

学号：************

******

指导老师：吴艳秋（讲师）

完成时间：2014年5月

目录

摘要 ............................................................................................................................................................... I Abstract........................................................................................................................................................ I I

1 引言 (1)

2 不定积分的求解方法 (1)

2.1 基本公式法 (1)

2.2 分项积分法、因式分解法 (2)

2.3 “凑”微分法（第一类换元积分法） (3)

2.4第二类换元积分法 (4)

2.5分部积分法 (4)

2.6有理函数的积分 (5)

3 各种方法所对应的题型 (5)

3.1 基本公式法 (5)

3.2 分项积分法、因式分解法 (6)

3.3 “凑”微分法（第一类换元积分法） (7)

3.4第二类换元积分法 (8)

3.5分部积分法 (8)

3.6有理函数的积分 (9)

4 解决不定积分的一般步骤 (10)

致谢 (11)

参考文献 (11)

归结不定积分的求解方法

林相群

（重庆三峡学院数学与统计学院数学与应用数学专业2010级重庆万州 404000）

摘要：不定积分的求解方法在本科阶段可以归为六大类：基本公式法、分项积分法+因式分解法、“凑”微分法（第一类换元积分法）、第二类换元积分法、分部积分法、有理函数的积分法。当我们看到所求不定积分已经对应了公式表中的某一条时，我们便用“公式法”求解。但实际问题一般较为复杂，所以我们都需将原题通过其他方法进行变换，使其满足公式再计算。“分项积分法+因式分解法”通过把多项式分解成单项式求积分，但结合三角恒等式，我们可以将高次三角函数降幂，化成容易积分的形式。当被积函数为复合函数时，我们多考虑换元积分法。“第一类换元积分法”通过为复合函数的中间变量“凑微分”达到解题目的。“第二类换元积分法”多用于当第一类无法实行时，但“第二类换元积分法”的换元形式比较不容易看出来，真正做到灵活运用需要累积许多经验。当被积函数是幂函数、三角函数、指数函数、对数函数中任意两个的乘积时，我们多考虑用“分部积分法”。“分部积分法”有着明显特征，并十分容易上手，是一种很好的解题方法。而“有理函数的积分法”与“第二类换元积分法”一样，没有特别固定的套路，多凭借经验和灵活运用。所以一般拿到题目可先考虑用别的方法。在拿到不定积分的题目时，我们要分析题目属于上述六种解题类型的哪一类。排除掉不可能的类型，再在可能的类型中进行进一步筛选，直到留下两种或两种以下的解题方法后，再进行尝试。若用某种方法解题时，无论怎么解都解不出答案，那么可先检查自己有没有运算的错误，或者是否选错了方法。总之，不定积分虽然有很多题型，但是解题的方法离不开上述六种，只要掌握了上述六种任何不定积分都不再是难题！

关键词：不定积分；基本公式法；换元积分法；分部积分法；有理函数的积分法

The method of calculating the indefinite integral

LIN Xiang-qun

(Grade 2007, Mathematics and Applied Mathematics, College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou, Chongqing 404000 )

Abstract:The method of indefinite integral in the undergraduate stage can be classified into six categories: basic formula method, component integration method & factorization method, "collect" differential method (the first kind of change of variable in an indefinite integral), the second kind of change of variable in an indefinite integral, integration by parts method, primitives of rational functions method. When we see for indefinite integral has corresponding formula in the table, we use "formula method". But the actual problem is more complicated, so we all shall transfer the indefinite integral through other methods to make it meet the formula in the end. "component integration method & factorization method" using for the polynomial into monomial then find indefinite integral respectively, and combined with trigonometric identity, we can handle high time trigonometric function to drop power, thus easy to integrate. When the integrand is composite function, we consider changing the variable. "The first kind of change the variable" working by the given the middle variable of the composite function to solving the problem. "The second kind of change of variable in an indefinite integral" is working when the first kind of failing to solve the problem. But "the second" is less likely to see immediately because that question is truly flexible and need to accumulate many experiences. When the integrand is mixing by power function, trigonometric function, exponential function and logarithmic function of any two, we consider using the "integration by parts method "."Integration by parts method” has obvious characteristic and is very easy to use, is a kind of good method to solve problems. And "primitives of rational functions method" is similar to "the second kind of change of variable in an indefinite integral" method: there is no special characteristic, all we need is more experiences and flexible insights. So we can consider to use other methods first when we get the problem, then analysis the kind of the six types to find which type is not helpful and which is until leaves one or two possible methods for further trying. If no matter how to solve the question all was fail, we can check if the first operation is error or if we choose the wrong way. All in all, indefinite integral question although have many type, the problem solving method is not far away from these method above. As long as to master the six kinds, any indefinite integral is no longer a problem!

Keywords: Indefinite Integral; Basic Formula Method; Change the Variable; Integration by Parts;

Primitives of Rational Functions