Improper Integral

- precal

Improper Integral

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- precal

|dw:1350067985984:dw|

- precal

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

- experimentX

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

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## More answers

- anonymous

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

- precal

|dw:1350068067914:dw|

- anonymous

yes

- anonymous

are you familiar with last three special integrals formula

- precal

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

- anonymous

did you use double substitution

- experimentX

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

- precal

|dw:1350068248916:dw|these two?

- experimentX

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

- precal

ok I will love to see the shortcut

- experimentX

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

- anonymous

@experimentX me to please

- experimentX

|dw:1350068353134:dw|

- anonymous

@experimentX nice one

- precal

I see it is the first one

- 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

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

- experimentX

hmm ... it really was improper intgral!!

- precal

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

- 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

this doesn't converge.

- precal

ok I am following you so far

- 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

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

- experimentX

|dw:1350068915623:dw|

- precal

|dw:1350069081160:dw|

- 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

@precal I'm not sure what you are asking

- precal

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

- myininaya

It is improper integral because of that infinity thing.

- experimentX

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

- precal

|dw:1350069479305:dw|
provided the limit exists

- precal

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

- precal

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

- experimentX

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

- 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

|dw:1350069792525:dw|

- precal

|dw:1350069861584:dw|

- 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

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

- experimentX

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

- precal

|dw:1350070363244:dw|

- precal

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

- experimentX

|dw:1350070434570:dw|

- 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

try watching this
http://www.youtube.com/watch?v=KhwQKE_tld0

- precal

ok thanks, I can use all the help

- precal

Thanks, I enjoyed that lecture.

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