1. INTEGRATION & TECHNIQUES OF INTEGRATION
CHAPTER 4
INTEGRATION
&
TECHNIQUES OF INTEGRATION

2. INTRODUCTION
The integration is the process of finding the definite or indefinite integral.
The inverse process of differentiation.
Evaluating the indefinite integral
Which is equivalent to finding a function F such as and C is arbitrary constant.
Evaluating the definite integral

3. INTRODUCTION Basic integration formula: (Refer textbook pg: 100-102)
Properties of Integral
Let f and g be the function of x and k is a constant, then:

4. INTRODUCTION
Example 1 Evaluate the following integral:

5. INTRODUCTION
Exercise 1 Evaluate the following integral:

6. INTRODUCTION
Example 2 Evaluate the following integral:

7. INTRODUCTION
Exercise 2 Evaluate the following integral:

8. TECHNIQUES OF INTEGRATION
There are three techniques of integration:
The Substitution Rule
Integration by Parts
Partial Fraction

9. THE SUBSTITUTION RULE
Theorem If is a differentiable function whose range is an interval I and f is continuous on I, then

10. THE SUBSTITUTION RULE Method: Substitute Calculate Substitute and
Evaluate
Replace

11. THE SUBSTITUTION RULE
Example 3 Using substitution rule, evaluate:

12. THE SUBSTITUTION RULE
Question (a) Identify u, hence find the derivative of u: Substitute u and du Substitute

13. THE SUBSTITUTION RULE
Exercise 3

14. INTEGRATION BY PARTS
A technique for simplifying integrals of the form:
Formula

15. INTRODUCTION
Example 4 Using integration by part, evaluate:

16. INTRODUCTION
Solution (a) Identify u and dv Substitute into the formula of IVP

17. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Consider a function,
where P and Q are polynomials and
Conditions
The degree of P must be less than degree of Q
If degree of P greater than Q, apply Long Division
Know the factors of Q. (Linear factors or Quadratic factors)

18. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Method of Partial Fraction for Proper function
A linear factor gives a partial fractions
A quadratic factor gives a partial fractions

19. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Example 5 Solve the following functions:

20. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Solution (a) The partial fraction has the form: Find the undetermined coefficients, A and B Let , then Let , then

21. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Therefore partial fraction has the form: Solve the integration:

22. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Example 6 Solve the following functions:

23. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Solution (a) The partial fraction has the form: Find the undetermined coefficients, A, B, C and D Expand and equate the coefficients:

24. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Solve simultaneously: Therefore : Integrate:

25. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Method of Partial Fraction for Improper function Apply Long Division

26. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTION
Example 7 Solve the following functions:

27. EXTRA EXERCISE
Solve the following functions: