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Math Help - Integration Problem

  1. #1
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    Integration Problem

    Is there any way to integrate the following?:

    \int \frac{dx}{\sqrt{a^{-1}-x^{-1}}}

    (Where "a" is a constant.)

    Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by GarmGarf View Post
    Is there any way to integrate the following?:

    \int \frac{dx}{\sqrt{a^{-1}-x^{-1}}}

    (Where "a" is a constant.)

    Any help would be appreciated.
    Ok that is the same with

    \int \frac{dx}{\sqrt{1/a-1/x}}}

    I hope this helps
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  3. #3
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    Quote Originally Posted by GarmGarf View Post
    Is there any way to integrate the following?:

    \int \frac{dx}{\sqrt{a^{-1}-x^{-1}}}

    (Where "a" is a constant.)

    Any help would be appreciated.

    Fist we would need to simplfy to get

    \frac{1}{\sqrt{\frac{1}{a}-\frac{1}{x}}}=\frac{1}{\sqrt{\frac{x-a}{ax}}}=\frac{\sqrt{ax}}{\sqrt{x-a}}

    So we have the integral

    \sqrt{a}\int \frac{\sqrt{x}}{\sqrt{x-a}}dx

    Now we let x=a\sec^2(t) \implies dx=2a\sec(t)[\sec(t)\tan(t)]dt

    \sqrt{a}\int \frac{\sqrt{x}}{x-a}dx=\sqrt{a}\int \frac{\sqrt{a}\sec(t)}{\sqrt{a\sec^2(t)-a}}(2a\sec^2(t)\tan(t)dt)=

    2a^{\frac{3}{2}}\int \sec^{3}(t)dt

    Here is how you integrate \sec^{3}(t)

    Integral of secant cubed - Wikipedia, the free encyclopedia

    So we get

    2a^{3/2}\left[\frac{1}{2}\sec(t)\tan(t)+\frac{1}{2}\ln|\sec(t)+\  tan(t)| \right]

    From here we just need to back substitue

    x=a\sec^2(t) \iff \frac{x}{a}=\sec^2(t) \iff \frac{x}{a}=\tan^2(t)+1

    \tan^2(t)=\frac{x-a}{a} \iff \tan(t)=\sqrt{\frac{x-a}{a}} and of course

    \sec(t) =\sqrt{\frac{x}{a}}

    I will leave the rest to you
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