1. A rational function is by definition the quotient of two polynomial functions, which has the following form: §4 有理函数的积分 Integration of Rational Function

2. the rational function is an proper fraction ； the rational function is an improper fraction. If there is no common factor between numerator and denominator Using the polynomial division, the improper fraction can change to the sum of a polynomial and proper fraction. Eg.

3. Difficulty: Chang the proper fraction to the sum of fractions. General rules for changing the proper fraction to the sum of fractions ： （ 1 ） If the denominator has a factor ， then disassemble Especially ： disassemble as

4. （ 2 ） If the denominator has a factor ，, then disassemble Especially ： disassemble as

5. Change the proper fraction to the sum of fractions – unknown coefficient( 待定系数法 ) Eg.1

6. Using the certain value to get let using A, B’s value Eg.2

7. Eg.3 get

8. Eg.4 Evaluate Sol.

9. Eg.5 Evaluate Sol.

10. Eg.5 Evaluate Sol. let

12. Eg.7 Eg.8

14. Eg.9 Evaluate Sol. 原式 注意本题技巧 本题用常规方法解很繁

15. 按常规方法解 第一步 令 比较系数定 a, b, c, d. 得 第二步 化为部分分式. 即令 比较系数定 A, B, C, D. 第三步 分项积分. 此解法较繁 !

16. Def.of t rigonometric function rational expression: 二、 The indefinite integral of trigonometric function rational expression the finite number of operations of constant and trigonometric function,write as

17. let （万能置换公式）

18. Eg.10 Evaluate Sol.From the formula

20. Sol.( 二 )

21. Eg.11 Evaluate Sol. （一）

22. Sol. （二）

23. Sol. （三） Tips Comparing the upper methods, you’ll realize the formula method is not the optimization, using other method first, if you can not evaluate the indefinite integral, then think about the formula method.

24. Eg.12 Evaluate Sol.

26. Type: Methods: Get rid of the radical expression. Eg.13 Evaluate Sol. Let 三、 The indefinite integral of simple irrational function

28. Eg.14 Evaluate Sol. Let Tips by using the least common multiple of the root exponent to get rid of the radical expression.

29. Eg.15 Evaluate Sol. First rationalize the denominator

30. 1. 可积函数的特殊类型 有理函数 分解 多项式及部分分式之和 三角函数有理式 万能代换 简单无理函数 三角代换 根式代换 2. 特殊类型的积分按上述方法虽然可以积出, 但不 要注意综合使用基本积分法, 简便计算. 一定简便, 内容小结

31. 练习题

34. 练习题答案