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Math Help - Improper Integrals (and one that isn't!)

  1. #1
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    Improper Integrals (and one that isn't!)

    1) \int_{0}^{1}\frac{1}{x*ln{x}}dx

    2) \int_{0}^{\infty}\frac{x}{e^{x}}dx

    3) \int_{0}^{\infty}\frac{1}{x*(ln{x})^{2}}dx

    These beasts are the only thing I've yet to conquer on my latest AP Calc BC take home test. I've thought about integration by parts and even simple u-substitution but I'm drawing a blank here. I'm somewhat sure I need to substitute in a value for infinity and take the limit as that value goes to infinity for the impropers, but I'm not really sure.

    Any help is appreciated. :>
    Last edited by zerot; November 27th 2007 at 04:55 PM.
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  2. #2
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    Let's do the second one.

    \int_{0}^{\infty}\frac{x}{e^{x}}dx

    \int_{0}^{L}\frac{x}{e^{x}}dx

    =\frac{-L}{e^{L}}-\frac{1}{e^{L}}+1

    \lim_{L\rightarrow{L}}\left[\frac{-L}{e^{L}}-\frac{1}{e^{L}}+1\right]=\boxed{1}

    As you should be able to see, as L is unbounded, we are left with 1
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  3. #3
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    Thanks, I thought it was something along those lines. I also edited my original post to make it more read-friendly.

    Also, would you mind elaborating more on how you found the anti-derivative of \int_{0}^{L}\frac{x}{e^{x}}dx? Sorry, I just don't see it. :<
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  4. #4
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    Hello, zerot!

    1)\;\;\int_{0}^{1}\frac{1}{x\ln x}dx
    We have: . \int\frac{1}{\ln x}\cdot\frac{dx}{x}

    Let: . u \:=\:\ln x\quad\Rightarrow\quad du \:=\:\frac{dx}{x}

    Substitute: . \int\frac{1}{u}\,du . . . . Got it?



    2)\;\;\int_{0}^{\infty}\frac{x}{e^x}dx
    This one requires by-parts . . .

    \begin{array}{ccccccc} u & = & x & \quad & dv & = & e^{-x}dx \\<br />
du & = & dx & \quad & v & = & -e^{-x} \end{array}

    We have: . -xe^{-x} + \int e^{-x}dx . . . . Carry on!



    3)\;\;\int_{0}^{\infty}\frac{1}{x(\ln x)^{2}}dx
    This is the same the first problem . . .

    We have: . \int\frac{1}{(\ln x)^2}\,\frac{dx}{x}

    Let: . u \:=\:\ln x\quad\Rightarrow\quad du \:=\:\frac{dx}{x}

    Substitute: . \int\frac{1}{u^2}\,du \;=\;\int u^{-2}du . . . . Okay?

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  5. #5
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    The problem is that I cannot evaluate the definite integral. I did the first and third problems the way you also did, but that leaves me finding ln|0| and ln|ln|0|| and that's what's hanging me up.
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