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Thread: difficult integral

  1. #1
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    difficult integral

    Hi everyone,

    Can you tell me if this is correct, please?

    Evaluate the following integral:

    x/(1+x^4 )dx

    u=x/(1+x^4)
    du=arctanx dx

    x/1+x^4=
    udu=
    (x/(1+x^4))arctanx+c

    Thank you very much
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chocolatelover View Post
    Hi everyone,

    Can you tell me if this is correct, please?

    Evaluate the following integral:

    x/(1+x^4 )dx

    u=x/(1+x^4)
    du=arctanx dx

    x/1+x^4=
    udu=
    (x/(1+x^4))arctanx+c

    Thank you very much
    i'm afraid that's incorrect. here's one way:


    $\displaystyle \int \frac {x}{1 + x^4}~dx = \int \frac {x}{1 + \left( x^2 \right)^2}~dx$

    We proceed by substitution

    Let $\displaystyle u = x^2$

    $\displaystyle \Rightarrow du = 2x~dx$

    $\displaystyle \Rightarrow \frac {1}{2} du = x~dx$

    So our integral becomes:

    $\displaystyle \frac {1}{2} \int \frac {1}{1 + u^2}~du$

    $\displaystyle = \frac {1}{2} \arctan u + C$

    $\displaystyle = \frac {1}{2} \arctan \left(x^2 \right) + C$




    Allow me to stress that $\displaystyle \int \frac {1}{1 + x^2}~dx = \arctan x + C$ NOT $\displaystyle \frac {d}{dx} \left( \frac {1}{1 + x^2} \right) = \arctan x $

    and you didn't even have that, you had $\displaystyle \frac {d}{dx} \left( \frac {x}{1 + x^4}\right) = \arctan x$ now that's just wrong
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  3. #3
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    $\displaystyle \int {\frac{x}
    {{1 + x^4 }}\,dx} = \frac{1}
    {2}\int {\frac{{\left( {x^2 } \right)'}}
    {{1 + \left( {x^2 } \right)^2 }}\,dx} = \frac{1}
    {2}\arctan x^2 + k,\,k\in\mathbb R$

    Jhevon, don't be so hard, not necessary to rub her(his) in, the "not"
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Krizalid View Post
    Jhevon, don't be so hard, not necessary to rub her(his) in, the "not"
    it wasn't my intention to be harsh.

    but i bet he/she will never forget now

    it's just tough love
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  5. #5
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    Thank you very much

    Regards
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