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Thread: find or evaluate the integral

  1. #1
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    find or evaluate the integral

    Hi

    $\displaystyle \displystyle \int \frac{1}{1 + \sqrt{x}}~dx $

    I have tried multiplying the numerator with $\displaystyle +\sqrt{x}$ and $\displaystyle -\sqrt{x}$ to factor out the denominator and got stuck
    I have tried substituted $\displaystyle 1 + \sqrt{x}$ and got stuck
    I have tried substituting $\displaystyle \sqrt{x}$ with u and replaced the numerator with $\displaystyle u - \sqrt{x}$ and came up with nothing...

    Insights please.
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  2. #2
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by ugkwan View Post
    Hi

    $\displaystyle \displystyle \int \frac{1}{1 + \sqrt{x}}~dx $

    I have tried multiplying the numerator with $\displaystyle +\sqrt{x}$ and $\displaystyle -\sqrt{x}$ to factor out the denominator and got stuck
    I have tried substituted $\displaystyle 1 + \sqrt{x}$ and got stuck
    I have tried substituting$\displaystyle \sqrt{x}$ with u and replaced the numerator with$\displaystyle u - \sqrt{x}$ and came up with nothing...

    Insights please.


    use $\displaystyle u = \sqrt {x} $ so $\displaystyle du = \frac {1}{2\sqrt{x}} $

    so now your integral will be

    $\displaystyle \displaystyle 2\int \frac {u}{u+1} du = \displaystyle 2 \int (1 - \frac {1}{1+u} ) du $



    can you continue from here ?
    Last edited by yeKciM; Sep 25th 2010 at 09:38 AM.
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  3. #3
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    so what is the antiderivative of $\displaystyle \frac {u}{u+1}$?
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  4. #4
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    nevermind my previous question. I figured out that by adding 1 and minusing 1 to the numerator, i can split the function, and then proceed with the antiderivative process.
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