Okay, I have a fairly stupid question.
How do I check wheter integral
exists?
When I have
, I have to check that f is bounded and continuous almost everywhere. Do I have to do the same when I have ? Is it enough to do the same?
Thank you.
Okay, I have a fairly stupid question.
How do I check wheter integral
exists?
When I have
, I have to check that f is bounded and continuous almost everywhere. Do I have to do the same when I have ? Is it enough to do the same?
Thank you.
If you are computing , then the additional requirement of existence is that the following two limits exist and are finite: and , where c is some constant that is convenient for your function.
(Summed up from Wikipedia article on Improper Integrals)
If either integral on the right side diverges, then we say that
diverges.
Let f be continuous on some interval, say, [a,b]. With the exception of at
some point c satisfying a<c<b. f(x) becomes infinite as x approaches c from
the left or right. If the two improper integrals
or
both converge, then we say that the improper integral
converges. That way we define:
See what I mean?. Is that what you were getting at?.
You can also say
The thing is, check for convergence or divergence of the limit.
Take for instance.
The other side can be shown to be the same and we have as the solution.
But, we split the integral at x=0. We did not have to do that. We could have done it anywhere and not affected the convergence or divergence.
Does that help a wee bit?.