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Integral x^2/(1+x^2)

  1. May 26, 2015 #1
    I would like to know the step by step solution of this integral:

    ∫ dx x^2 /( 1+x^2)

    I tried to solve it integrating by parts with u = x dv =x /(1+x^2) , or with hyperbolic functions, but I always get stuck...

    Thank you
  2. jcsd
  3. May 26, 2015 #2
    Can you show us where you get stuck?
  4. May 26, 2015 #3
    ∫ dx x^2 /( 1+x^2)

    u = x dv =x /(1+x^2)


    ∫ dx x^2 /( 1+x^2) = x log(1+x^2) -∫ dx log(1+x^2) ...stop

    I also tried

    u = x^2 dv = 1 /(1+x^2)


    ∫ dx x^2 /( 1+x^2) = x^2 atan(x) -∫ dx 2x atan(x) ...stop
  5. May 26, 2015 #4


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    Try ##x = tan(u)##
  6. May 26, 2015 #5


    Staff: Mentor

    The integrand is an improper rational expression (degree of numerator = degree of denominator).

    You can either use polynomial long division to get a proper rational expression, or do the following, which is easier:
    $$\int \frac{x^2~dx}{1 + x^2} = \int \frac{1 + x^2 - 1~dx}{1 + x^2}$$
    Now split into two integrals, one of which is trivial and the other you probably already know.
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