1. Mathematics

2. Session Indefinite Integrals - 3

3. Session Objectives  Three Standard Integrals  Integrals of the form  Integration Through Partial Fractions  Class Exercise

4. Three Standard Integrals

5. Integrals of the Form Reduce the given integral to one of the following forms:

6. Example-1

7. Example - 2

8. Solution Cont.

9. Example - 3

10. Integrals of the Form We use the following method: (ii) Obtain the values of A and B by comparing the coefficients of like powers of x. Then the integral reduces to

11. Example - 4

12. Solution Cont.

14. Integrals of the Form We use the following method: (iii) Now, we evaluate the integral by the method discussed earlier.

15. Example - 5

16. Solution Cont.

18. Integration Through Partial Fractions (Type – 1) When denominator is non-repeated linear factors where A, B, C are constants and can be calculated by equating the numerator on RHS to numerator on LHS and then substituting x = a, b, c,... or by comparing the coefficients of like powers of x.

19. Example - 6

20. Solution Cont.

21. Type - 2 When denominator is repeated linear factors where A, B, C, D, E and F are constants and value of the constants are calculated by substitution as in method (1) and remaining are obtained by comparing coefficients of equal powers of x on both sides.

22. Example - 7

23. Solution Cont.

24. Type - 3 When denominator is non-repeated quadratic factors where A, B, C are constants and are determined by either comparing coefficients of similar powers of x or as mentioned in method 1.

25. Example - 8

26. Solution Cont.

27. Type - 4 When denominator is repeated quadratic factors where A, B, C, D, E and F are constants and are determined by equating the like powers of x on both sides or giving values to x. Note: If a rational function contains only even powers of x, then we follow the following method: (i)Substitute x 2 = t (ii)Resolve into partial fractions (iii)Replace t by x 2

28. Example – 9

29. Solution Cont.

30. Example - 10 Solution: Here degree of N r > degree of D r.

31. Solution Cont.

32. Thank you