Your SlideShare is downloading. ×
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Calculus II - 1
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

Calculus II - 1

1,922

Published on

Stewart Calculus Section 7.1

Stewart Calculus Section 7.1

Published in: Education, Technology
0 Comments
3 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total Views
1,922
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
94
Comments
0
Likes
3
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • \n
  • 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 ( ) ( ) = ( )· ( ) ( ) ( )

    ×