Checking whether integral exists

• Oct 24th 2008, 03:34 PM
marianne
Checking whether integral exists
Okay, I have a fairly stupid question.

How do I check wheter integral
$\int_{-\infty}^{\infty} f(x) dx$ exists?
When I have
$\int_{a}^{b}f(x)dx$, I have to check that f is bounded and continuous almost everywhere. Do I have to do the same when I have $<-\infty, \infty>$? Is it enough to do the same?

Thank you.
• Oct 24th 2008, 04:15 PM
Jameson
If you are computing $\int_{- \infty}^{\infty} f(x)dx$, then the additional requirement of existence is that the following two limits exist and are finite: $\lim_{a \rightarrow -\infty} \int_{a}^{c} f(x)dx$ and $\lim_{b \rightarrow \infty} \int_{c}^{b} f(x)dx$, where c is some constant that is convenient for your function.

(Summed up from Wikipedia article on Improper Integrals)
• Oct 24th 2008, 04:17 PM
galactus
$\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{0}f(x)dx+\int_{0}^{\infty}f(x)dx$

If either integral on the right side diverges, then we say that

$\int_{-\infty}^{\infty}f(x)dx$ 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

$\int_{a}^{c}f(x)dx$ or $\int_{c}^{b}f(x)dx$

both converge, then we say that the improper integral

$\int_{a}^{b}f(x)dx$ converges. That way we define:

$\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b} f(x)dx$

See what I mean?. Is that what you were getting at?.

You can also say $\int_{a}^{b}f(x)dx=\lim_{L\to b^{-}}\int_{a}^{L}f(x)dx$

$\int_{a}^{+\infty}f(x)dx=\lim_{L\to {+\infty}}\int_{a}^{L}f(x)dx$

The thing is, check for convergence or divergence of the limit.

Take $\int_{-\infty}^{\infty}\frac{1}{1+x^{2}}dx$ for instance.

$\lim_{L\to {+\infty}}\int_{0}^{L}\frac{1}{1+x^{2}}dx$

$=\lim_{L\to {+\infty}}\left[tan^{-1}(x)\right]_{0}^{L}$

$=\lim_{l\to {+\infty}}tan^{-1}(L)=\frac{\pi}{2}$

The other side can be shown to be the same and we have ${\pi}$ 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?.
• Oct 24th 2008, 04:19 PM
toraj58
$\int_{-\infty}^{\infty} f(x) dx$

is equal with:

$\lim_{a\to\infty}\int_{-a}^c f(x) dx + \lim_{a\to\infty}\int_c^a f(x) dx$
• Oct 25th 2008, 01:25 AM
marianne
Thank you all, you've been most helpful!
I think I got it. :-) The example galactus provided really sorted it out for me.