1. DIFFERENTIATION & INTEGRATION
Chapter 4
DIFFERENTIATION & INTEGRATION

2. OUTLINE OF CHAPTER 4: DIFFERENTIATION & INTEGRATION
Integration by Substitution
Integration by Parts
Integration by Tabular Method
Integration by Partial Fraction
Differentiation
Derivative of power function
(power rule)
Derivative of a constant times a function
Derivative of sum and difference rules
Product rule
Quotient rule
Derivative of trigonometric functions
Derivative of exponential and logarithmic functions
Chain rule
Implicit Differentiation

3. 4.2.1 DERIVATIVE of power rule
Example 4.1:
Exercise 4.1:

4. 4.2.2 DERIVATIVE of a constant times a function
Example 4.2:
Exercise 4.2:

5. 4.2.3 DERIVATIVE of sum & difference rules
Example 4.3:
Exercise 4.3:

6. Exercise 1: power rule

7. Exercise 1: answer

8. 4.2.4 the product rules
Example 4.4:
Exercise 4.4:

9. Exercise 2: product rule

10. Exercise 2: answer

11. 4.2.5 the quotient rules
Example 4.5:
Exercise 4.5:

12. Exercise 3: quotient rule

13. Exercise 3: answer

14. Exercise 4:

15. 4.2.6 DERIVATIVE of trigonometric functions

16. Example 4.6:
Exercise 4.6:

17. Exercise 5: trigonometric functions

18. 4.2.7 DERIVATIVE of logarithmic functions

19. Example 4.7:
Exercise 4.7:

20. Properties of ln:

21. 4.2.8 the chain rules
Example 4.8:
Exercise 4.8:

22. Example: chain rule

23. Exercise 6:

24. Exercise 6: answer

25. Conclusion (differentiation)
Power Rule
Product Rule
Quotient Rule
Chain Rule

26. 4.2.9 implicit differentiation
Implicit differentiation is the process of taking the derivative when y is defined implicitly or in others y is a function of x.
STEP 1: Differentiate both side with respect to x.
STEP 2: Collect dy/dx terms on the left hand side of the
equation
STEP 3: Solve for dy/dx

27. Example 4.9:
Exercise 4.9:

28. Exercise 7:

29. Exercise 8

31. 4.3 integration
Integration is the inverse process of differentiation process.
Derivative formula
Equivalent integral formula

32. Exercise 9:

33. 4.3.1 indefinite integral
The constant factor k can be taken out from an integral,
The integral of a sum or difference equals the sum or difference of the integral, that is

34. Example 4.12:
Exercise 4.11:

35. Exercise 10:

36. 4.3.2 definite integral
If f(x) is a real-valued continuous function on closed interval [a,b] and F(x) is an indefinite integral of f(x) on [a,b], then

37. Basic Properties of Definite Integrals:

38. Example 4.13:
Exercise 4.12:

39. Exercise 11:

40. 4.4 technique of integration
Integration by Substitution
Integration by Parts
Integration by Tabular Method
Integration by Partial Fraction

41. 4.4.1 integration by substitution
Step 1: Choose appropriate u, Step 2: Compute Step 3: Substitute and in the integral Step 4: Evaluate the integral in term of u Step 5: Replace , so that the final answer will be in term of x

42. Example:

44. Example 4.14:
Exercise 4.13:

45. Example:

47. Exercise 12:

48. 4.4.2 integration by parts
Involve products of algebraic and transcendental functions. For example:
The formula:
A priority order to choose u:

49. 4.4.2 integration by parts
Step 1: Choose the appropriate u and dv. (Note: the expression dv must contain dx) Step 2: Differentiate u to obtain du and integrate dv to obtain v. (Note: Do not include the constant C when integrating dv since we are still in the process of integrating) Step 3: Substitute u, du, v and dv into formula and complete the integration. (Note: Remember to include the constant C in the final answer)

50. Example 4.15:
Exercise 4.13:

51. Exercise 13:

52. 4.4.3 integration by tabular methods
The formula: Note: Can be used to evaluate complex integrations especially repeated integrations (when u=xn). Step 1: u can be differentiated repeatedly with respect to x until becoming zero. Step 2: v’ can be integrated repeatedly with respect to x.

53. Example:

54. Exercise 14:

55. 4.4.4 integration by partial fraction
Consider a function, where P and Q are polynomials and Q(x)≠ 0
Case I (improper fraction):
If the deg P(x) ≥ deg Q(x), long division is applied to obtain
remainder
Example:

56. Exercise 15:

57. Case II (proper fraction): If the deg P(x) < deg Q(x), factorize the denominator (Q(x)) into its prime factors. i) Linear factor ii) Linear factor

58. Case II (proper fraction): If the deg R(x) < deg T(x), factorize the denominator (T(x)) into its prime factors. iii) Quadratic factor iv) Quadratic factor

59. Example 4.16:
Exercise 4.15:

60. Example:

62. Example:

64. Exercise 16:

65. Exercise: