1. Chapter 6 6.4 Integration of substitution and integration by parts of the definite integral

2. As we all known, the integration of substitution ( or “change of variables”) and integration by parts are very important tools to evaluate the indefinite integrals. In this section we extend these methods to the definite integrals. 1. Integration of substitution At first, the substitution technique extends to definite integrals

3. Theorem 1 Theorem 1 (Substitution in a definite integral) Substitution formula of definite integrals

4. Proof

5. Notice One important is the necessary change in the limits of integration. The following examples will apply this technique. Example 1 Solution

6. Example 2 Solution Additivity over intervals

7. Change of variables Newton-Leibniz formula Example 3 Solution

8. Example 4 Solution

9. Example 5

10. Solution

11. 2. Important simplification formulas Theorem 2

12. Solution

14. Notice The properties of definite integrals for odd and even functions provide a easy way to evaluate their definite integrals. The following examples will apply the property. Example 6 Solution even function

15. odd function Area of unit circle Example 7

16. Solution odd function Example 8

17. Solution

18. Thus, we choice ( D ) Theorem 3 Proof

19. f (x+T)=f (x) The integral properties of periodic functions provide also a easy way to evaluate some definite integrals. Next example will apply the property. Example 9

20. Solution

21. Proof （ 1 ） Let Example 10

24. 3 Integration by parts Theorem 4 Next, the integration by parts extends to definite integrals Integration by parts for definite integrals

25. Proof This completes the proof

26. Solution Example 11

27. Example 12 Solution

28. Example 13 Solution

29. Example 14 Solution A

30. Thus, we choose (A)(A)

31. Example 15 Solution

33. Example 16 Solution