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Math Help - Integration

  1. #1
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    Integration

    I can't figure out why when integrating

    <br /> <br />
-4{x}^\frac{3}{2}<br /> <br />

    you get

    <br /> <br />
-\displaystyle{\frac{8}{3}{x}}^\frac{3}{2}<br /> <br />

    and not

     <br />
-8{x}^\frac{3}{2}<br /> <br />

    Why does it become a fraction?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Envy View Post
    I can't figure out why when integrating

    <br /> <br />
-4{x} \frac{3}{2}<br /> <br />

    you get

    <br /> <br />
-\displaystyle{\frac{8}{3}{x}}<br />
\frac{3}{2}<br /> <br />

    and not

     <br />
-8{x} \frac{3}{2}<br /> <br />

    Why does it become a fraction?
    The rule is:

    \int k x^r\ dx= \frac{k}{r+1}x^{r+1} +C

    In your case k=-4, r=3/2 and so:

    \int -4 x^{3/2}\ dx= -\ \frac{4}{(5/2)}x^{5/2} +C=-\ \frac{8}{5}x^{5/2} + C

    CB
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  3. #3
    MHF Contributor
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    Quote Originally Posted by Envy View Post
    I can't figure out why when integrating

    <br /> <br />
-4{x}^\frac{3}{2}<br /> <br />

    you get

    <br /> <br />
-\displaystyle{\frac{8}{3}{x}}^\frac{3}{2}<br /> <br />

    and not

     <br />
-8{x}^\frac{3}{2}<br /> <br />

    Why does it become a fraction?
    You shouldn't get either of those answers you've listed...

    If f(x) = ax^n then it's antiderivative is F(x) = \int{ax^n\,dx} = \frac{ax^{n + 1}}{n + 1}+C.

    In other words, you add 1 to your power, and then divide by the new power.


    In your case you have -4x^{\frac{3}{2}}.


    Add 1 to the power and you get \frac{5}{2}.


    Divide by your new power and you get

    -4\div\frac{5}{2}

     = -4\times\frac{2}{5}

     = -\frac{8}{5}.



    So your antiderivative is -\frac{8x^{\frac{5}{2}}}{5} + C where C is a constant.
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post


    -\ \frac{4}{(5/2)}x^{5/2} +C=-\ \frac{8}{5}x^{5/2} + C
    Thanks, I forgot to flip the fraction and then times it

    Quote Originally Posted by Prove It View Post
    You shouldn't get either of those answers you've listed...
    Totally correct, I wrote down the wrong quesion as -4{x}^\frac{3}{2} which was my partially intergrated answer using {ax^{n + 1}}, the actual question was -4{x}^\frac{1}{2}!
    Last edited by Envy; June 3rd 2009 at 03:05 AM.
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