## precal Group Title Improper Integral one year ago one year ago

1. precal Group Title

|dw:1350067985984:dw|

2. precal Group Title

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

3. experimentX Group Title

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

4. ipm1988 Group Title

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

5. precal Group Title

|dw:1350068067914:dw|

6. ipm1988 Group Title

yes

7. ipm1988 Group Title

are you familiar with last three special integrals formula

8. precal Group Title

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

9. ipm1988 Group Title

did you use double substitution

10. experimentX Group Title

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

11. precal Group Title

|dw:1350068248916:dw|these two?

12. experimentX Group Title

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

13. precal Group Title

ok I will love to see the shortcut

14. experimentX Group Title

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

15. ipm1988 Group Title

16. experimentX Group Title

|dw:1350068353134:dw|

17. ipm1988 Group Title

@experimentX nice one

18. precal Group Title

I see it is the first one

19. experimentX Group Title

|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 Group Title

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

21. experimentX Group Title

hmm ... it really was improper intgral!!

22. precal Group Title

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

23. myininaya Group Title

|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 Group Title

this doesn't converge.

25. precal Group Title

ok I am following you so far

26. precal Group Title

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 Group Title

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

28. experimentX Group Title

|dw:1350068915623:dw|

29. precal Group Title

|dw:1350069081160:dw|

30. precal Group Title

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 Group Title

@precal I'm not sure what you are asking

32. precal Group Title

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

33. myininaya Group Title

It is improper integral because of that infinity thing.

34. experimentX Group Title

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

35. precal Group Title

|dw:1350069479305:dw| provided the limit exists

36. precal Group Title

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

37. precal Group Title

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

38. experimentX Group Title

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

39. precal Group Title

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 Group Title

|dw:1350069792525:dw|

41. precal Group Title

|dw:1350069861584:dw|

42. experimentX Group Title

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 Group Title

|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 Group Title

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

45. precal Group Title

|dw:1350070363244:dw|

46. precal Group Title

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

47. experimentX Group Title

|dw:1350070434570:dw|

48. precal Group Title

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 Group Title