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# Calculus II - 1

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Stewart Calculus Section 7.1

Stewart Calculus Section 7.1

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• ### Transcript

• 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 ( ) ( ) = ( )· ( ) ( ) ( )