1. THE INDEFINITE INTEGRAL
Chapter 5
THE INDEFINITE INTEGRAL

2. New Words Rational function 有理函数 Partial fraction 部分分式 Polynomial 多项式
Factorization 因式分解式
Proper fraction 真分式 Factor 分解因式
Improper fraction 假分式
Reducible 可约的 Denominator 分母
Irreducible 不可约的 Numerator 分子

3. 5.4 Integration of certain elementary functions
1 How to integrate certain rational functions
At first, we will rewrite any rational function in a much simpler form

4. (1) The form of rational function
(2) The properties of rational function
a. Any rational improper fraction can be expressed as
the sum of a polynomial and a rational proper fraction.

5. b. Any polynomial function Q(x) with real coefficients can factorize into the product of first-degree polynomial and the second-degree irreducible polynomials in R

6. The general procedure for representing a proper rational function consists of four steps:

8. (4) Integrate a rational function
Next, we focus on integrating the four types of function I, II, III, IV. Once we know how to integrate them, we can integrate any rational function

12. Example 1
Solution
Since the degree of the numerator is greater than the degree of the denominator, it is improper. Divide:

15. Remark:

16. The above method is called comparison of coefficients
The above method is called comparison of coefficients. It depends on the fact that if two polynomials are equal, then their corresponding coefficients are equal.
The next example illustrates a way for finding the partial fractions when the factorization of the denominator involves only linear factors and none of them is repeated.
Example 2
Solution

19. If a linear factor is repeated you may use either substitution or comparison coefficients.
Example 3
Solution

21. Example 4
Solution
Clearing the denominator gives

23. Remark
(1) Either the substitution method or the comparison of coefficients method are used in the above example.
(2) But we can also use other methods to integrate
rational functions.

24. Example 5
Solution
Example 6
Solution

25. 2. How to integrate rational trigonometric functions
So far in this chapter you have met three techniques for computing integrals. The first, integration of substitution

26. and the second, integration by parts, are used most often
and the second, integration by parts, are used most often. Partial fractions applies only to a special class of integrands, the rational functions. In this section we compute some rational trigonometric functions.

27. Example 7
Solution

28. Example 8
Solution

29. Example 9
Solution

32. A half-angle substitution

33. Example 10
Solution

35. 3. How to integrate simple irrational functions
Some simple irrational functions can be transformed into a rational function by proper substitution.
For example:

37. Example 11
Solution

39. Example 12
Solution

40. See you next time