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Math Help - substitution problem

  1. #1
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    substitution problem

    hey i've got a question i'm finding really tricky. if anyone can put this in latex i'd be really thankful!

    use the substitution x-p = 1/u to show the integral

    dx/(x - p)sqrt[(x - p)(x - q)] equals


    2/(q - p) . sqrt[(x - q)/(x - p)]

    thank you!
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  2. #2
    MHF Contributor red_dog's Avatar
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    I=\int\frac{dx}{(x-p)\sqrt{(x-p)(x-q)}}=\frac{2}{q-p}\sqrt{\frac{x-q}{x-p}}

    x-p=\frac{1}{u}\Rightarrow dx=-\frac{1}{u^2}du

    I=\int\frac{-\frac{1}{u^2}du}{\frac{1}{u}\sqrt{\frac{1}{u}\left  (\frac{1}{u}+p-q\right)}}=

    =-\int\frac{du}{\sqrt{1+u(p-q)}}=\frac{2}{q-p}\sqrt{1+u(p-q)}+C

    Now replace u with \frac{1}{x-p}.
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  3. #3
    Member Mauritzvdworm's Avatar
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    do you mean you need to show the following
    \int \frac{dx}{(x-p)\sqrt{(x-p)(x-q)}}=\frac{2}{p-q}\sqrt{\frac{x-q}{x-p}}
    using the substitution x-p=\frac{1}{u}??
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