precal
  • precal
Improper Integral
Mathematics
katieb
  • katieb
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precal
  • precal
|dw:1350067985984:dw|
precal
  • precal
I have the solution, looking for help to understand the process
experimentX
  • experimentX
lol .. "Improper Integral" is not quite the term you describe it.

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anonymous
  • anonymous
break x^4 in (x^2)^2 then use substitution
precal
  • precal
|dw:1350068067914:dw|
anonymous
  • anonymous
yes
anonymous
  • anonymous
are you familiar with last three special integrals formula
precal
  • precal
three special integral formula? not sure which one you are referring to
anonymous
  • anonymous
did you use double substitution
experimentX
  • experimentX
let x^2 = u sub ... this would end up into inverse typerbolic function.
precal
  • precal
|dw:1350068248916:dw|these two?
experimentX
  • experimentX
well ... you can try that. but I'll give you a shortcut.
precal
  • precal
ok I will love to see the shortcut
experimentX
  • experimentX
look at the the inverse hyperbolic of sine http://en.wikipedia.org/wiki/Inverse_hyperbolic_function
anonymous
  • anonymous
@experimentX me to please
experimentX
  • experimentX
|dw:1350068353134:dw|
anonymous
  • anonymous
@experimentX nice one
precal
  • precal
I see it is the first one
experimentX
  • 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.
precal
  • precal
|dw:1350068751083:dw| solution manual has this as a second step
experimentX
  • experimentX
hmm ... it really was improper intgral!!
precal
  • precal
Is this because of the square root function's domain being continuous on [a, infinity)?
myininaya
  • 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 \[2x dx=\sec^2( \theta) d \theta \]
experimentX
  • experimentX
this doesn't converge.
precal
  • precal
ok I am following you so far
precal
  • 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)
myininaya
  • 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.
experimentX
  • experimentX
|dw:1350068915623:dw|
precal
  • precal
|dw:1350069081160:dw|
precal
  • 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
myininaya
  • myininaya
@precal I'm not sure what you are asking
precal
  • precal
In the definition of Improper integrals over infinite intervals I have 3 definitions
myininaya
  • myininaya
It is improper integral because of that infinity thing.
experimentX
  • experimentX
probably with definition of improper integral ... when you have limit goes to infinity ... you have improper integral of first kind.
precal
  • precal
|dw:1350069479305:dw| provided the limit exists
precal
  • precal
I should mention that if f is continuous on [a,infinity), then for the above first definition
precal
  • precal
2nd definition If f is continuous on (-infinty, b], then|dw:1350069615971:dw| provided the limit exists.
experimentX
  • experimentX
|dw:1350069732672:dw| this is called improper integral of First Kind.
precal
  • precal
3rd definition If f is continuous on (-infinity, infinity), then |dw:1350069715530:dw| probided both limits exist, where c is any real number
experimentX
  • experimentX
|dw:1350069792525:dw|
precal
  • precal
|dw:1350069861584:dw|
experimentX
  • 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.
precal
  • precal
|dw:1350070134924:dw|maybe it is because it should have stated the following limits, I think I see a typo from the source
experimentX
  • experimentX
the second kind appears when you have singularity in domain.|dw:1350070237179:dw|
precal
  • precal
|dw:1350070363244:dw|
precal
  • precal
now I see the connection, it has to be establish - no wonder you made the statement earlier about it not being improper
experimentX
  • experimentX
|dw:1350070434570:dw|
precal
  • 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
experimentX
  • experimentX
try watching this http://www.youtube.com/watch?v=KhwQKE_tld0
precal
  • precal
ok thanks, I can use all the help
precal
  • precal
Thanks, I enjoyed that lecture.

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