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Math Help - integration help

  1. #1
    Newbie yuriythebest's Avatar
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    integration help

    right why does this happen? can someone explain what is going on?

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  2. #2
    Super Member
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    First, you used 2 to multiply by 1: \frac{1}{2} \int \frac{2}{x^2(x^2+2)}

    Then, you used x^2 to add 0: \frac{1}{2} \int \frac{(2+x^2)-x^2}{x^2(x^2+2)}

    Then, you split (2+x^2) and x^2: \frac{1}{2} \int \frac{2+x^2}{x^2(2+x^2)}~dx - \frac{1}{2} \int \frac{x^2}{x^2(x^2+2)}~dx

    And as you can see in the end, all this work was an elaborate plan to achieve a cancellation and split it into two easier integrals: \frac{1}{2} \int \frac{1}{x^2}~dx - \frac{1}{2} \int \frac{1}{x^2+2}~dx

    First integral should be simple enough for you, and as for the second integral, you should see that by taking 2 out as a factor and doing some manipulation, you'll get a standard arctan form.
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  3. #3
    Newbie yuriythebest's Avatar
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    Quote Originally Posted by Chop Suey View Post
    First, you used 2 to multiply by 1: \frac{1}{2} \int \frac{2}{x^2(x^2+2)}

    Then, you used x^2 to add 0: \frac{1}{2} \int \frac{(2+x^2)-x^2}{x^2(x^2+2)}

    Then, you split (2+x^2) and x^2: \frac{1}{2} \int \frac{2+x^2}{x^2(2+x^2)}~dx - \frac{1}{2} \int \frac{x^2}{x^2(x^2+2)}~dx

    And as you can see in the end, all this work was an elaborate plan to achieve a cancellation and get a more simpler integral: \frac{1}{2} \int \frac{1}{x^2}~dx - \frac{1}{2} \int \frac{1}{x^2+2}~dx

    First integral should be simple enough for you, and as for the second integral, you should see that by taking 2 common factor and doing some manipulation, you'll get a standard arctan form.
    I am having great difficulty understanding your formulas- can you write them down traditionally?

    EDIT: nvm now they all of a sudden look good
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  4. #4
    Super Member Showcase_22's Avatar
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    ln x does not apply here since the differential of the denominator does not appear at the top.

    The method that you have posted appears to be a partial fraction method.
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