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Math Help - Improper Integral Help

  1. #1
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    Improper Integral Help

    I am just starting on the improper integral section. I am having trouble with this problem because I cannot seem to break it down easily into something that does not have (x-5) in the denominator. If I try to do lim R -> 5+, I will still get an undefined answer. I am doing something wrong, but I do not know what.

    12. \int^6_5 \frac{dx}{(x-5)^{3/2}}

    I get the general antiderivative, but I cannot get rid of the denominator, and since the numerator is just 1, I cannot use L'Hopital's Rule.
    Thanks for your help.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Using the substitution t=x-5


    \displaystyle\int_5^6\dfrac{dx}{(x-5)^{3/2}}=\displaystyle\int_0^1\dfrac{dt}{t^{3/2}}=\displaystyle\lim_{ \epsilon \to 0^+}{\displaystyle\int_{\epsilon}^{ 1}\dfrac{dt}{t^{3/2}}}=\displaystyle\lim_{ \epsilon \to 0^+}\left[-\dfrac{2}{\sqrt{t}}\right]_{\epsilon}^1=\ldots
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  3. #3
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    Thank you very much!
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Warrenx View Post
    Thank you very much!
    You are welcome!
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