Must Improper integral, from -inf to inf, be broken down into 2 separate limits?

Mr_Random_Guy

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Must the integral from -inf to inf be broken down into two separate limits as follows:
(lim s --> -inf of the integral from s to 0) + (lim t --> +inf of the integral from 0 to t)

as opposed to just:
lim n --> inf of the integral from -n to n
 
Must the integral from -inf to inf be broken down into two separate limits as follows:
(lim s --> -inf of the integral from s to 0) + (lim t --> +inf of the integral from 0 to t)

as opposed to just:
lim n --> inf of the integral from -n to n

Yes.

There are cases where the latter might converge, but the former does not. The former is the definition of the improper integral.
 
Must the integral from -inf to inf be broken down into two separate limits as follows:
(lim s --> -inf of the integral from s to 0) + (lim t --> +inf of the integral from 0 to t)

as opposed to just:
lim n --> inf of the integral from -n to n
Yes, but there is no need to break it up at 0, as you can break it up at 7 or pi or 3.567 or a where a is some real number.
 
Yes.

There are cases where the latter might converge, but the former does not. The former is the definition of the improper integral.
Can you please show us an example or two. Better yet, can you please show us how to generate these problems where you can't go from -n to n. Thanks!
 
Can you please show us an example or two. Better yet, can you please show us how to generate these problems where you can't go from -n to n. Thanks!
One example would be where the interval includes a vertical assymptote.
 
Can you please show us an example or two. Better yet, can you please show us how to generate these problems where you can't go from -n to n. Thanks!

Here's a simple example: integrate the inverse tangent from -infinity to infinity. What happens when you use -n to n? What happens when you break it up?

I originally had in mind a slightly different situation, but this is the easiest to come up with. The idea is that the two ends can cancel one another out (in one sense or another) when taken together, so that you don't get to see that each of them diverges on its own.
 
We can make it even simpler: \(\displaystyle \int_{-\infty}^{\infty} x dx\). Try it out!
 
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