1. CHAPTER 4 INTEGRATION

2. Integration is the process inverse of differentiation process. The integration process is used to find the area of region under the curve. INDEFINITE INTEGRAL

3.  Examples :

4. EXERCISE 4

5.  Examples : THE DEFINITE INTEGRAL

6.  Integration by Substitution – Change of Variables  Reversing the “chain rule” (from differentiation).  General integration by substitution :  Integration by Substitution steps:  Figure out the “inner” function; call it u(x).  Compute..  Replace all expressions involving the variable x and dx with the new variable u and du. Use the differential formula to replace the differential dx.  Evaluate the resulting “u” integral. If you can’t evaluate the integral, try a different choice of u.  Replace all occurrences of the variable u in the antiderivative with the appropriate function of x. TECHNIQUES OF INTEGRATION

7.  Example 1 : Calculate

8.  Example 2: Find

9. EXERCISE 5

11.  Integration by Parts  Example 3 :Find

12. EXERCISE 6

13.  Partial Fractions A quotient of polynomials : can be expressed as a simpler fraction called partial fractions. This technique is used for rewriting problems so that they can integrate. For example, the integral can be rewritten as using the method partial fractions. Then it is easily integrated as

14.  Types of Partial Fraction 1.Improper Fraction

15. 2.Proper Fraction

17.  Example 4: (Case 1)

18.  Example 5 :(Improper Fraction)

19. EXERCISE 7