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# Calculus II - 1

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Stewart Calculus Section 7.1

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• ### Calculus II - 1

1. 1. Calculus IIBook: Stewart CalculusContent:Integration TechniquesAdvanced Integration TopicsDifferential EquationsParametric Equations and Polar CoordinatesSeries and Taylor ExpansionsVectors and Vector Calculus
2. 2. Prerequisite
3. 3. PrerequisiteIdeas of Derivative and Integral.
4. 4. PrerequisiteIdeas of Derivative and Integral.The relationship and difference betweenDefinite and Indefinite Integral.
5. 5. PrerequisiteIdeas of Derivative and Integral.The relationship and difference betweenDefinite and Indefinite Integral.Product and quotient rules, chain rule.
6. 6. PrerequisiteIdeas of Derivative and Integral.The relationship and difference betweenDefinite and Indefinite Integral.Product and quotient rules, chain rule.Fundamental Theorem of Calculus.
7. 7. PrerequisiteIdeas of Derivative and Integral.The relationship and difference betweenDefinite and Indefinite Integral.Product and quotient rules, chain rule.Fundamental Theorem of Calculus.The derivative and integral of the mostimportant functions: trigonometric function,exponential function, power function...
8. 8. 7.1 Integration by Parts
9. 9. 7.1 Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )
10. 10. 7.1 Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )Integrate both sides by indefinite integral: ( )· ( )= ( ) ( ) + ( ) ( )
11. 11. 7.1 Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )Integrate both sides by indefinite integral: ( )· ( )= ( ) ( ) + ( ) ( )Formula of Integration by Parts: ( ) ( ) = ( )· ( ) ( ) ( )
12. 12. 7.1 Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )Integrate both sides by indefinite integral: ( )· ( )= ( ) ( ) + ( ) ( )Formula of Integration by Parts: ( ) ( ) = ( )· ( ) ( ) ( )
13. 13. ( ) ( ) = ( )· ( ) ( ) ( )
14. 14. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )
15. 15. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Let ( )= , ( )= ( )
16. 16. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Let ( )= , ( )= ( )then ( )= , ( )= ( )
17. 17. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Let ( )= , ( )= ( )then ( )= , ( )= ( )so · ( ) = ·( ( )) ·( ( ))
18. 18. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Let ( )= , ( )= ( )then ( )= , ( )= ( )so · ( ) = ·( ( )) ·( ( )) = · ( )+ ( )+
19. 19. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )
20. 20. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Why not ( )= , ( )= ( ) ?
21. 21. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Why not ( )= , ( )= ( ) ?then ( )= , ( )= ( )
22. 22. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Why not ( )= , ( )= ( ) ?then ( )= , ( )= ( )so · ( ) = · ( ) · ( )
23. 23. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Why not ( )= , ( )= ( ) ? ( )= , ( )= ( ) Even morethen co mplicate d !so · ( ) = · ( ) · ( )
24. 24. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find · ( )Why not ( )= , ( )= ( ) ? ( )= , ( )= ( ) Even morethen co mplicate d !so · ( ) = · ( ) · ( )Conclusion: in general only one way works.We want to make sure the integrand is simpler.
25. 25. ( ) ( ) = ( )· ( ) ( ) ( )
26. 26. ( ) ( ) = ( )· ( ) ( ) ( )
27. 27. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find
28. 28. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=
29. 29. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()=
30. 30. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()=If we choose ()= , ()=
31. 31. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()=If we choose ()= , ()=then ()= , ()=
32. 32. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()= Simpler!If we choose ()= , ()=then ()= , ()=
33. 33. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()=
34. 34. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()=so =
35. 35. ( ) ( ) = ( )· ( ) ( ) ( )Ex: FindLet ()= , ()=then ()= , ()= New problem !so =
36. 36. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find =
37. 37. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find =Now let ()= , ()=
38. 38. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find =Now let ()= , ()=then ()= , ()=
39. 39. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find =Now let ()= , ()=then ()= , ()=therefore = = +
40. 40. ( ) ( ) = ( )· ( ) ( ) ( )Ex: Find =Now let ()= , ()=then ()= , ()=therefore = = +so = + +
41. 41. Integration by Parts
42. 42. Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )
43. 43. Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )Integrate both sides by definite integral: ( )· ( ) = ( ) ( )+ ( ) ( )
44. 44. Integration by PartsProduct Rule of derivative: [ ( ) · ( )] = ( ) ( ) + ( ) ( )Integrate both sides by definite integral: ( )· ( ) = ( ) ( )+ ( ) ( )Formula of Integration by Parts: ( ) ( ) = ( )· ( ) ( ) ( )
45. 45. ( ) ( ) = ( )· ( ) ( ) ( )
46. 46. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: Find
47. 47. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: FindLet ( )= , ( )=
48. 48. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: FindLet ( )= , ( )=then ( )= , ( )=
49. 49. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: FindLet ( )= , ( )=then ( )= , ( )= / /so =
50. 50. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: FindLet ( )= , ( )=then ( )= , ( )= / /so = / =
51. 51. ( ) ( ) = ( )· ( ) ( ) ( ) /Ex: Find =Let ( )= , ( )=then ( )= , ( )= / /so = / =
52. 52. Integration by Parts Indefinite Integral ( ) ( ) = ( )· ( ) ( ) ( ) Definite Integral ( ) ( ) = ( )· ( ) ( ) ( )