Stewart Calculus Section 7.1

1. Calculus II
Book: Stewart Calculus
Content:
Integration Techniques
Advanced Integration Topics
Differential Equations
Parametric Equations and Polar Coordinates
Series and Taylor Expansions
Vectors and Vector Calculus

3. Prerequisite

4. Prerequisite
Ideas of Derivative and Integral.

5. Prerequisite
Ideas of Derivative and Integral.
The relationship and difference between
Definite and Indefinite Integral.

6. Prerequisite
Ideas of Derivative and Integral.
The relationship and difference between
Definite and Indefinite Integral.
Product and quotient rules, chain rule.

7. Prerequisite
Ideas of Derivative and Integral.
The relationship and difference between
Definite and Indefinite Integral.
Product and quotient rules, chain rule.
Fundamental Theorem of Calculus.

8. Prerequisite
Ideas of Derivative and Integral.
The relationship and difference between
Definite and Indefinite Integral.
Product and quotient rules, chain rule.
Fundamental Theorem of Calculus.
The derivative and integral of the most
important functions: trigonometric function,
exponential function, power function...

10. 7.1 Integration by Parts

11. 7.1 Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )

12. 7.1 Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )
Integrate both sides by indefinite integral:
( )· ( )= ( ) ( ) + ( ) ( )

13. 7.1 Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )
Integrate both sides by indefinite integral:
( )· ( )= ( ) ( ) + ( ) ( )
Formula of Integration by Parts:
( ) ( ) = ( )· ( ) ( ) ( )

14. 7.1 Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )
Integrate both sides by indefinite integral:
( )· ( )= ( ) ( ) + ( ) ( )
Formula of Integration by Parts:
( ) ( ) = ( )· ( ) ( ) ( )

15. ( ) ( ) = ( )· ( ) ( ) ( )

16. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )

17. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Let ( )= , ( )= ( )

18. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Let ( )= , ( )= ( )
then ( )= , ( )= ( )

19. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Let ( )= , ( )= ( )
then ( )= , ( )= ( )
so · ( ) = ·( ( )) ·( ( ))

20. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Let ( )= , ( )= ( )
then ( )= , ( )= ( )
so · ( ) = ·( ( )) ·( ( ))
= · ( )+ ( )+

21. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )

22. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Why not ( )= , ( )= ( ) ?

23. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Why not ( )= , ( )= ( ) ?
then ( )= , ( )= ( )

24. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Why not ( )= , ( )= ( ) ?
then ( )= , ( )= ( )
so · ( ) = · ( ) · ( )

25. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Why not ( )= , ( )= ( ) ?
( )= , ( )= ( ) Even more
then
co mplicate d
!
so · ( ) = · ( ) · ( )

26. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find · ( )
Why not ( )= , ( )= ( ) ?
( )= , ( )= ( ) Even more
then
co mplicate d
!
so · ( ) = · ( ) · ( )
Conclusion: in general only one way works.
We want to make sure the integrand is simpler.

27. ( ) ( ) = ( )· ( ) ( ) ( )

28. ( ) ( ) = ( )· ( ) ( ) ( )

29. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find

30. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=

31. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=

32. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=
If we choose ()= , ()=

33. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=
If we choose ()= , ()=
then ()= , ()=

34. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()= Simpler!
If we choose ()= , ()=
then ()= , ()=

35. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=

36. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=
so =

37. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find
Let ()= , ()=
then ()= , ()=
New problem
!
so =

38. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find =

39. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find =
Now let ()= , ()=

40. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find =
Now let ()= , ()=
then ()= , ()=

41. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find =
Now let ()= , ()=
then ()= , ()=
therefore = = +

42. ( ) ( ) = ( )· ( ) ( ) ( )
Ex: Find =
Now let ()= , ()=
then ()= , ()=
therefore = = +
so = + +

44. Integration by Parts

45. Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )

46. Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )
Integrate both sides by definite integral:
( )· ( ) = ( ) ( )+ ( ) ( )

47. Integration by Parts
Product Rule of derivative:
[ ( ) · ( )] = ( ) ( ) + ( ) ( )
Integrate both sides by definite integral:
( )· ( ) = ( ) ( )+ ( ) ( )
Formula of Integration by Parts:
( ) ( ) = ( )· ( ) ( ) ( )

48. ( ) ( ) = ( )· ( ) ( ) ( )

49. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find

50. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find
Let ( )= , ( )=

51. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find
Let ( )= , ( )=
then ( )= , ( )=

52. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find
Let ( )= , ( )=
then ( )= , ( )=
/ /
so =

53. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find
Let ( )= , ( )=
then ( )= , ( )=
/ /
so =
/
=

54. ( ) ( ) = ( )· ( ) ( ) ( )
/
Ex: Find =
Let ( )= , ( )=
then ( )= , ( )=
/ /
so =
/
=

56. Integration by Parts
Indefinite Integral
( ) ( ) = ( )· ( ) ( ) ( )
Definite Integral
( ) ( ) = ( )· ( ) ( ) ( )