1. 反微分與不定積分 及其性質 1. 反微分 (Antiderivatives) 2. 不定積分 (Indefinite Integral) 3. 積分規則 (Rules of Integration) 4. 替代法 (Substitution) page 358~379

2. Definition of Antiderivative : If F’(x)= f(x), then F(x) is an antiderivative of f (x)

3. Example EX1(a) F(x)=10x, F’(x)=10  F(x) is an antiderivative of f(x)= 10

4. Example EX1(b) F(x)=, F’(x)=2x  F(x) is an antiderivative of f(x)= 2x

5. Example EX2 Find an antiderivative of f(x)=,  is an antiderivative of When n=5, is an antiderivative of  An antiderivative of is

6. Example EX3 Find an antiderivative of f(x)= let F(x)= and F’(x)= =f(x)  F(x) is an Antiderivative of f(x)

7. Notice F(x)= and F’(x)= let G(x)= +2 and G’(x)= =f(x) let H(x)= +100 and H’(x)= =f(x)  F(x), G(x) and H(x) are antiderivative of f(x) Antiderivatives of f(x) differ by a constant

8. Property of Antiderivatives If F(x) and G(x) are both antiderivatives of a function f (x) on an interval, then there is a constant C such that F(x)-G(x)=C The arbitrary real number C is called an integration constant ( 積分常數 ) EX: F(x)=2x+2 and F’(x)=2 G(x)=2x+100 and G’(x)=2 H(x)=2x+10000 and H’(x)=2

9. Indefinite Integral The family of all antiderivative of f(x) (F’(x) = f(x)) is indicated by where C : Integral Constant : Integral Sign( 積分符號 ) f(x) : Integrand( 積分函數 ) dx : integral of f(x) with respect to x is called an Indefinite Integral

10. Power Rule For any real number, EX4 Use the power rule to find each indefinite integral. (a) (b) (c) (d)

11. Constant Multiple Rule and Sum or Difference Rule If all indicated integrals exist, and for any real number a, b

12. Example: Find because are constants. Then we can use C to represent integral constant where

13. EX5 Use the rules to find each integral. (a) (by constant multiple rule) (by power rule) (b)

14. EX6 Use the rules to find each integral (c)

15. EX5 Use the rules to find each integral. (a) (by constant multiple rule) (by power rule) (b)

16. Review of Derivative of Exponential Function f (x) =  f’ (x) =

17. Indefinite Integrals of Exponential Functions,,

18. Review of Derivative of Exponential Function f (x) =  f’ (x) =

19. Indefinite Integrals of Exponential Functions,,

20. EX7 Exponential Functions (a) (b) (c) (d)

21. Indefinite Integrals of Exponential Functions,,

22. EX7 Exponential Functions (a) (b) (c) (d)

23. Indefinite Integral of Derivative of Logarithmic Function f (x) = where  f’ (x) =

24. EX8 Integrals (a) (b)

25. EX9 Cost-1 Suppose a publishing company has found that the marginal cost at a level of production of x thousand books is given by and that the fixed cost (the cost before the firsr book can be produced) is $25,000. Find the cost function

26. EX9 Cost-2 and use the indefinite integral rules to integrate the function When x=0, C(0)=25,000, K=25,000 The cost function is

27. Review of the Chain Rule 1. Let u = g(x), =g’(x) 2. Let w = f(g(x))=f(u), =f’(u) 3.  f(g(x)) = f’(u) g’(x)=f’(g(x))g’(x)

28. Substitution Rules If u=g(x) is a differential function where du=g’(x)dx, then EX: 1., du=2xdx 2.

29. General Power Rule for Integrals For u=f(x) and du=f’(x)dx, EX1: Find let u =, du =6x 

30. EX2 General Power Rule Find Let u =,

31. EX3 General Power Rule Find Let u =,

32. Indefinite Integrals of For u=f(x) and du=f’(x)dx, Indefinite Integrals of For u=f(x) and du=f’(x)dx,

33. Example of Substitution EX4: Find let u =, du = 

34. Example of Substitution EX5: Find let u =, 

35. EX6 Substitution Find Let u =,, x =1- u

36. EX Integrals Find Let

37. Example: Find Let

38. EX Integrals Find Let g(x)=sin x

39. Example: Find

41. It ’ s not suitable to apply the substitution rules

42. Substitution Method The choice of u is one of the following: 1. The quantity under a root or raised to a power 2. The exponent on e 3. The quantity in the denominator