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Math Help - Is something amiss with this integral?

  1. #1
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    Is something amiss with this integral?

    I just got back from a Calc II test.

    There was one question which seemed bizarre; I copied it down and searched the textbook but couldn't find a match.

    It's the integral from [0,2] of x^2 + x^-1 + 3^x. (We solve by the Fundamental Theorem of Calculus).

    Am I correct in saying that the integral of x^-1 is ln|x|?

    If I am, then the resulting evaluation for 0 gives me an undefined answer, correct?
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  2. #2
    MHF Contributor Amer's Avatar
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    \int_{0}^{2}x^2+x^{-1}+3^{x} dx

    \int_{0}^{2}x^2+x^{-1}+3^{x} dx = \frac{x^3}{3} + lnx + \frac{3^x}{ln3}

    it is correct that the integrate of x^-1 is ln(x) take absolute value of x

    the other thing is the integrate of 3^x

    the integrate of e^x = e^x ln(e) but since ln(e)= 1 so we didn't mention it

    that's all
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  3. #3
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    The integral \int_0^2 {\left( {x^2  + x^{ - 1}  + 3^x } \right)dx} is an improper integral because x^{-1} is not defined at x=0.

    Have you studied improper integrals?
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  4. #4
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    Quote Originally Posted by Plato View Post
    Have you studied improper integrals?
    No, which makes me think my professor must have made a mistake.
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  5. #5
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    Quote Originally Posted by Chizum View Post
    No, which makes me think my professor must have made a mistake.
    Are you sure it was not \int_{\color{red}1}^2 {\left( {x^2  + x^{ - 1}  + 3^x } \right)dx} or maybe x^{\color{red}{-2}}?
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  6. #6
    Senior Member Spec's Avatar
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    Improper integrals were introduced in Calc I for me. I can't imagine there being a Calc II course were they don't teach you improper integrals, or at least have it as a prerequisite.
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  7. #7
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    Quote Originally Posted by Plato View Post
    Are you sure it was not \int_{\color{red}1}^2 {\left( {x^2  + x^{ - 1}  + 3^x } \right)dx} or maybe x^{\color{red}{-2}}?
    Pretty sure, I spent at least 5 minutes thinking about it and then copied it down onto a piece of scratch paper.

    @ Spec,

    The text we're using doesn't seem to introduce Improper Integrals till 8.8 (according to the index). We're on like 6.5 right now.
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