INTEGRATION & TECHNIQUES OF INTEGRATION CHAPTER 4 INTEGRATION & TECHNIQUES OF INTEGRATION

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

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:

INTRODUCTION Example 1 Evaluate the following integral:

INTRODUCTION Exercise 1 Evaluate the following integral:

INTRODUCTION Example 2 Evaluate the following integral:

INTRODUCTION Exercise 2 Evaluate the following integral:

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

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

THE SUBSTITUTION RULE Method: Substitute Calculate Substitute and Evaluate Replace

THE SUBSTITUTION RULE Example 3 Using substitution rule, evaluate:

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

THE SUBSTITUTION RULE Exercise 3

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

INTRODUCTION Example 4 Using integration by part, evaluate:

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

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)

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

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

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

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

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

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:

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

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

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

EXTRA EXERCISE Solve the following functions: