## precal 2 years ago Improper Integral

1. precal

|dw:1350067985984:dw|

2. precal

I have the solution, looking for help to understand the process

3. experimentX

lol .. "Improper Integral" is not quite the term you describe it.

4. ipm1988

break x^4 in (x^2)^2 then use substitution

5. precal

|dw:1350068067914:dw|

6. ipm1988

yes

7. ipm1988

are you familiar with last three special integrals formula

8. precal

three special integral formula? not sure which one you are referring to

9. ipm1988

did you use double substitution

10. experimentX

let x^2 = u sub ... this would end up into inverse typerbolic function.

11. precal

|dw:1350068248916:dw|these two?

12. experimentX

well ... you can try that. but I'll give you a shortcut.

13. precal

ok I will love to see the shortcut

14. experimentX

look at the the inverse hyperbolic of sine http://en.wikipedia.org/wiki/Inverse_hyperbolic_function

15. ipm1988

16. experimentX

|dw:1350068353134:dw|

17. ipm1988

@experimentX nice one

18. precal

I see it is the first one

19. experimentX

|dw:1350068400466:dw| probably you would get inverse hyperbolic for this types ... but generally formula is given as |dw:1350068503882:dw| ... lol all these forms are inverse hyperbolic function. people usually don't use these.

20. precal

|dw:1350068751083:dw| solution manual has this as a second step

21. experimentX

hmm ... it really was improper intgral!!

22. precal

Is this because of the square root function's domain being continuous on [a, infinity)?

23. myininaya

|dw:1350068740846:dw| Start with a right triangle Label the sides such that the inside of that radical you have above appears somewhere. |dw:1350068780160:dw| So we will use substitution $x^2=\tan( \theta)$ |dw:1350068798338:dw| Now we need to take derivative of both sides of our substitution $2x dx=\sec^2( \theta) d \theta$

24. experimentX

this doesn't converge.

25. precal

ok I am following you so far

26. precal

yes in the end, I know this diverges. I know the final answer, just trying to learn the process or understand the process (is a better description)

27. myininaya

$2x dx=\sec^2(\theta) d \theta = > x dx =\frac{1}{2} \sec^2(\theta) d \theta$ Now you go to your little expression and make your "suby's" happen.

28. experimentX

|dw:1350068915623:dw|

29. precal

|dw:1350069081160:dw|

30. precal

I guess I am not sure in the set up why [a, infinity) was used vs (-infinity, b] or (-infinity, infinity) is it because of the domain of the square root function

31. myininaya

@precal I'm not sure what you are asking

32. precal

In the definition of Improper integrals over infinite intervals I have 3 definitions

33. myininaya

It is improper integral because of that infinity thing.

34. experimentX

probably with definition of improper integral ... when you have limit goes to infinity ... you have improper integral of first kind.

35. precal

|dw:1350069479305:dw| provided the limit exists

36. precal

I should mention that if f is continuous on [a,infinity), then for the above first definition

37. precal

2nd definition If f is continuous on (-infinty, b], then|dw:1350069615971:dw| provided the limit exists.

38. experimentX

|dw:1350069732672:dw| this is called improper integral of First Kind.

39. precal

3rd definition If f is continuous on (-infinity, infinity), then |dw:1350069715530:dw| probided both limits exist, where c is any real number

40. experimentX

|dw:1350069792525:dw|

41. precal

|dw:1350069861584:dw|

42. experimentX

as long as you have infinity as limits ... just relax .. they all are of same type of improper integral. probably you are confused with improper integral of second kind.

43. precal

|dw:1350070134924:dw|maybe it is because it should have stated the following limits, I think I see a typo from the source

44. experimentX

the second kind appears when you have singularity in domain.|dw:1350070237179:dw|

45. precal

|dw:1350070363244:dw|

46. precal

now I see the connection, it has to be establish - no wonder you made the statement earlier about it not being improper

47. experimentX

|dw:1350070434570:dw|

48. precal

ok I thinkg I got it, it was the set up that threw me off. I think I understand the solution better now. Thanks

49. experimentX

50. precal

ok thanks, I can use all the help

51. precal

Thanks, I enjoyed that lecture.