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Math Help - Integration problem. Do they want me to use partial fractions?

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    Integration problem. Do they want me to use partial fractions?

    I am working on an assignment. The first task is:

    Find the indefinite integral of (x+2)/(x^2)+x

    I factored out an x of the denominator, and split to partial fractions that were easy to integrate. My result was: 2*ln|x| - ln|x+1|

    My questions are:
    Have i done it right?
    Even if i have, is there another (simpler, or more clever) way to do this?
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  2. #2
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    I assume you are actually trying to write

    \displaystyle{\int{\frac{x+2}{x^2+x}\,dx}}

    in which case, partial fractions is the way to go.
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    Junior Member pirateboy's Avatar
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    If by partial fractions you mean partial fraction decomposition, you don't even have to do that. You just have to see that
    \displaystyle \frac{x+2}{x^2 + 2} = \frac{x}{x^2+2}+\frac{2}{x^2+2}

    You'll then end up with two separate integrals you'll have to bust some sweet moves on. One should be straight up u-substitution and the other well...here's a BIG hint (it's not a bad idea to have this memorized)

    \displaystyle \int\frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C
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    Great, thanks!
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  5. #5
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    Quote Originally Posted by pirateboy View Post
    If by partial fractions you mean partial fraction decomposition, you don't even have to do that. You just have to see that
    \displaystyle \frac{x+2}{x^2 + 2} = \frac{x}{x^2+2}+\frac{2}{x^2+2}

    You'll then end up with two separate integrals you'll have to bust some sweet moves on. One should be straight up u-substitution and the other well...here's a BIG hint (it's not a bad idea to have this memorized)

    \displaystyle \int\frac{dx}{a^2+x^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C
    Except it's not \displaystyle{\frac{x + 2}{x^2 + 2}}, it's \displaystyle{\frac{x+2}{x^2 + x}}.
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