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Math Help - Basic Intergration Question

  1. #1
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    Basic Intergration Question

    Looking to integrate the following.

    \int \frac{x^4}{x^2 + 1} dx

    We usually have to re-write the denominator as a negative and solve that way yeah? For example...

    \int \frac{x^4 + 1}{x^2}

    is

    \int x^2 + x^-2 and integrate.

    How do I solve it though with the +1 on the denominator? This might be a really really stupid question, but I can't simplify..... can I?

    P.s. This is more of a problem about simplifying rather than integration I suspect.
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  2. #2
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    I'm not that great at calculus, but I noticed your problem and think I may know the answer. The key is in the following:

    x^4 = (x^2 + 1)(x^2 - 1) + 1

    Your integral is then:

    \int {\frac {(x^2 + 1)(x^2 - 1) + 1} {x^2 + 1}} dx = \int {x^2 - 1 + \frac {1}{x^2 + 1}} dx

    You should be right integrating that simplified form, but just in case you need the answer for reference, it's:

    \frac {x^3}{3} - x + arctan(x)
    Last edited by drew.walker; May 31st 2009 at 03:45 AM.
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  3. #3
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    Quote Originally Posted by drew.walker View Post
    I'm not that great at calculus, but I noticed your problem and think I may know the answer. The key is in the following:

    x^4 = (x^2 + 1)(x^2 - 1) + 1

    Your integral is then:

    \int {\frac {(x^2 + 1)(x^2 - 1) + 1} {x^2 + 1}} dx = \int {x^2 - 1 + \frac {1}{x^2 + 1}} dx

    You should be right integrating that simplified form, but just in case you need the answer for reference, it's:

    \frac {x^3}{3} - x + arctan(x)
    + C.
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