1. Mathematics

2. Session Indefinite Integrals -1

3. Session Objectives  Primitive or Antiderivative  Indefinite Integral  Standard Elementary Integrals  Fundamental Rules of Integration  Methods of Integration 1. Integration by Substitution, Integration Using Trigonometric Identities

4. Primitive or Antiderivative then the function F(x) is called a primitive or an antiderivative of a function f(x).

5. Cont. If a function f(x) possesses a primitive, then it possesses infinitely many primitives which can be expressed as F(x) + C, where C is an arbitrary constant.

6. Indefinite Integral Let f(x) be a function. Then collection of all its primitives is called indefinite integral of f(x) and is denoted by where F(x) + C is primitive of f(x) and C is an arbitrary constant known as ‘constant of integration’.

7. Cont. will have infinite number of values and hence it is called indefinite integral of f(x). If one integral of f(x) is F(x), then F(x) + C will be also an integral of f(x), where C is a constant.

8. Standard Elementary Integrals

9. Cont. The following formulas hold in their domain

10. Cont.

12. Fundamental Rules of Integration

13. Example - 1

14. Example - 2

15. Cont.

16. Example - 3

17. Example - 4

18. Integration by Substitution If g(x) is a differentiable function, then to evaluate integrals of the form We substitute g(x) = t and g’(x) dx = dt, then the given integral reduced to After evaluating this integral, we substitute back the value of t.

19. Cont.

20. Example - 5 Solution :

21. Integration Using Trigonometric Identities

22. Example - 6

23. Integration Using Trigonometric Identities

24. Example - 7 [Using 2sinAcosB = sin (A + B) + sin (A – B)]

25. Integration by Substitution

26. Example - 8

27. Solution Cont. Method - 2

28. Example - 9

29. Some Standard Results

30. Integration by Substitution

31. Example - 10

32. Integration by Substitution Use the following substitutions. (i) When power of sinx i.e. m is odd, put cos x = t, (ii) When power of cosx i.e. n is odd, put sinx = t, (iii) When m and n are both odd, put either sinx = t or cosx = t, (iv) When both m and n are even, use De’ Moivre’s theorem.

33. Example - 11 Powers of sin x and cos x are odd. Therefore, substitute sinx = t or cosx = t We should put cosx = t, because power of cosx is heigher

34. Cont.

35. Example - 12

36. Example - 13

37. Example - 14

38. Thank you