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Math Help - indefinite integral problem

  1. #1
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    indefinite integral problem

    I'm trying to find the indefinite integral of:

    (root)(x) / (root)(x) - 3 , the -3 is not under the root-sign

    I've tried various u-subsitutions, none of which have made the problem any easier. I also tried using polynomial division to simplify the expression to

    1 + ( 3 / ((root)(x) - 3), which I still cannot solve.

    Would appreciate any help!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: indefinite integral problem

    Using the substitution t=\sqrt{x} you'll obtain an easy integral (rational on t) .
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  3. #3
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    Re: indefinite integral problem

    Quote Originally Posted by gralla55 View Post
    I'm trying to find the indefinite integral of:

    (root)(x) / (root)(x) - 3 , the -3 is not under the root-sign

    I've tried various u-subsitutions, none of which have made the problem any easier. I also tried using polynomial division to simplify the expression to

    1 + ( 3 / ((root)(x) - 3), which I still cannot solve.

    Would appreciate any help!
    Is it \displaystyle \int{\frac{\sqrt{x}}{\sqrt{x} - 3}\,dx}? If so...

    \displaystyle \begin{align*}\int{\frac{\sqrt{x}}{\sqrt{x} - 3}\,dx} &= \int{\frac{x}{\sqrt{x}(\sqrt{x} - 3)}\,dx} \\ &= 2\int{\frac{x}{2\sqrt{x}(\sqrt{x} - 3)}\,dx}\\ &= 2\int{\frac{u^2}{u - 3}\,du}\textrm{ after making the substitution }u = \sqrt{x} \implies du = \frac{dx}{2\sqrt{x}} \end{align*}
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    Re: indefinite integral problem

    Thanks! Yes, that is the integral. Actually I tried doing just that, I just didn't know how to evaluate that integral either... Do you use long division to get:

    u + (3u / (u +3))

    ?

    Thanks again!
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    Re: indefinite integral problem

    Quote Originally Posted by gralla55 View Post
    Thanks! Yes, that is the integral. Actually I tried doing just that, I just didn't know how to evaluate that integral either... Do you use long division to get:

    u + (3u / (u +3))

    ?

    Thanks again!
    You're on the right track, but it should be \displaystyle u + \frac{3u}{u - 3}. You now need to long divide again.
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  6. #6
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    Re: indefinite integral problem

    Lol, you're right, 3u / (u-3) is of course correct. I used long division again and FINALLY got the right answer!

    Thank you so much!!
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