Methods of integration

It highlights the methods to the problems of Integration

1. Integration methods Dr. Pankaj Das Mail.id: [email protected]

2. Strategy of integration As we have seen, integration is more challenging than differentiation. In finding the derivative of a function, it is obvious which differentiation formula we should apply. However, it may not be obvious which technique we should use to integrate a given function. The main challenge is to recognize which technique or formula to use. A prerequisite for strategy selection is a knowledge of the basic integration formulas.

3. Strategy of integration Once armed with these basic integration formulas, you might try this strategy: Simplify the integrand if possible. Look for an obvious substitution. Classify the integrand according to its form. Try again.

4. Strategy of integration Sometimes, the use of algebraic manipulation or trigonometric identities will simplify the integrand and make the method of integration obvious. Here are some example

5. Indefinite integral: Example 1. Indefinite integral can be represent as

6. Example 2.

7. Example 3.

8. Definite integral D etermine the definite integral in each case as defined below. Example 1: Let our problem

9. Example 2. Let our problem

10. Manipulate The Integrand Algebraic manipulations (rationalizing the denominator, using trigonometric identities) may be useful in transforming the integral into an easier form. Here are an example

11. Manipulate The Integrand For instance, ∫ tan 2 x sec x dx is a challenging integral. If we make use of the identity tan 2 x = sec 2 x – 1, we can write : Then, if ∫ sec 3 x dx has previously been evaluated, that calculation can be used in the present problem.

12. Use Several Methods Sometimes, two or three methods are required to evaluate an integral. The evaluation could involve several successive substitutions of different types . It might even combine integration by parts with one or more substitutions.

13. Linearity

14. 2. Function of linear function This can be solved by two methods Inspection method Substitution method

15. Inspection method

16. Inspection method

17. Substitution method

18. Substitution method

19. Substitution method Algebraic manipulations (rationalizing the denominator, using trigonometric identities) may be useful in transforming the integral into an easier form. These manipulations may be more substantial than in Step 1 and may involve some ingenuity.

20. Some more examples Let our problem, We can write as, Example 1

21. Some more examples Substitute u = cos x: We can write as,

22. Integration by parts

23. Integration by parts

24. Integration by parts

25. Integration by parts

26. Integration by parts

27. Reduction formula Reduction formulae are the formulae that may be used repeatedly to express the integral of a complicated function in terms of simpler one . The reduction formulae are generally obtained by the application of the rule of integration by parts. Reduction formula is regarded as a method of integration. Integration by reduction formula helps to solve the powers of elementary functions, polynomials of arbitrary degree, products of transcendental functions and the functions that cannot be integrated easily, thus, easing the process of integration and its problems.

28. Formulas for Reduction in Integration The reduction formula can be applied to different functions including trigonometric functions like sin, cos, tan, etc., exponential functions, logarithmic functions, etc. Here, the formula for reduction is divided into 4 types: For exponential functions For trigonometric functions For inverse trigonometric functions For hyperbolic trigonometric functions For algebraic functions

29. Generating a reduction formula: Definite integral Integrands of the form and Using the integration by parts formula: it is easily shown that:

30. Generating a reduction formula: indefinite integral Writing: then can be written as: This is an example of a reduction formula .

31. Generating a reduction formula: indefinite integral Sometimes integration by parts has to be repeated to obtain the reduction formula. For example:

32. Generating a reduction formula: D efinite integral When the integral has limits the reduction formula may be simpler. For example:

33. Generating a reduction formula : Similarly we can derive the following forms, Similarly,

34. Generating a reduction formula :

35. Some important reduction formulas:

36. Example 1.

37. Thank you