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Math Help - prove this integral does not exist

  1. #1
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    prove this integral does not exist

    integral ( 1/x, x, 0,1)
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  2. #2
    GAMMA Mathematics
    colby2152's Avatar
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    Quote Originally Posted by szpengchao View Post
    integral ( 1/x, x, 0,1)
    Is that \int_0^1 \frac{1}{x} dx?

    If so, note it will not exist at the bound: x=0 because not only is the function not continuous at that point (cannot divide by zero), the resultant integral \ln(x) cannot take values less than or equal to zero.
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  3. #3
    MHF Contributor arbolis's Avatar
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    You can notice that \int_0^1 \frac{1}{x} dx=\int_1^\infty \frac{1}{x} dx.
    So we have \int_1^\infty \frac{1}{x} dx= limit when b tends to \infty of \int_1^b \frac{1}{x}dx = limit when b tends to \infty of ln(x) evaluated in b minus ln(x) evaluated in 1. Which is equal to limit when b tends to \infty of ln(x) evaluated in b, which is \infty, thus divergent.
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  4. #4
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    prove

    how can u get the first step
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  5. #5
    Math Engineering Student
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    It is a simple reciprocal substitution, try it with x=\frac1u.
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  6. #6
    MHF Contributor arbolis's Avatar
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    Ok. I said \int_0^1 \frac{1}{x} dx=\int_1^\infty \frac{1}{x} dx, this implies that the graph of f(x)=\frac{1}{x} is symmetric with respect to the graph of y=x. It means that if f(x)=\frac{1}{x}, then f^{-1}(x)=f(x). In other words, f has its own bijection equals to itself.
    Now to prove it, you just have to show that f(f^{-1}(x))=x.

    f(f^1(x)) is equals to \frac{1}{f^1(x)}=\frac{\frac{1}{1}}{\frac{1}{x}}=x. It is demonstrated. If we take a precise example maybe you'll understand better. If x=5, then f(5)=\frac{1}{5}. That means that to get 5 with our function, we need to evaluated it in \frac{1}{5}, which works. Maybe you will understand even better seeing the graph. It's trivial that the area under the curve of f(x)=\frac{1}{x} from 0 to 1 is equal to the area from 1 to \infty. To be rigorous, just say f is its own bijection (and show it as I did for you), that should be enough.
    Last edited by arbolis; May 24th 2008 at 12:17 PM.
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