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Stewart Calculus Section 7.1

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

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