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Math Help - Checking whether integral exists

  1. #1
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    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.
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  2. #2
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    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)
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  3. #3
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    \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?.
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  4. #4
    Junior Member toraj58's Avatar
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    \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
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  5. #5
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    Thank you all, you've been most helpful!
    I think I got it. :-) The example galactus provided really sorted it out for me.
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