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Math Help - Integrate this!

  1. #1
    Junior Member freestar's Avatar
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    Integrate this!

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

    Tried both u substitution letting u = 1+x^2, and u = atan(x), neither worked. Tried using integration by parts as well. Any suggestions in the right direction?
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  2. #2
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    e^(i*pi)'s Avatar
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    Re: Integrate this!

    Quote Originally Posted by freestar View Post
    \int \frac{x^2}{1+x^2}dx

    Tried both u substitution letting u = 1+x^2, and u = atan(x), neither worked. Tried using integration by parts as well. Any suggestions in the right direction?
    \dfrac{x^2}{1+x^2} = \dfrac{x^2+1-1}{1+x^2} = \dfrac{x^2+1}{1+x^2} - \dfrac{1}{1+x^2} = 1 - \dfrac{1}{1+x^2}

    \int \dfrac{x^2}{1+x^2}dx  = \int 1dx - \int \dfrac{1}{1+x^2}dx


    The second integral is a standard integral according to Wolfram (are you "allowed" to know this result or do you have to show/derive it?)
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  3. #3
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    Re: Integrate this!

    You can also use the substitution \tan u=x directly in the first integral. Then keep in mind that 1+\tan^2u=\sec^2u.
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  4. #4
    Junior Member freestar's Avatar
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    Re: Integrate this!

    Thank you sir! I am allowed to know this result. I'll keep this trick in mind for the future. Much appreciated.
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