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calculus

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evaluate the integral:

(x^3)/((x^2)+1)

  • calculus - ,

    For rational functions, first step is to see if the degree of the numerator is higher than that of the denominator. If this is the case (as in the present problem), do a long division to reduce the numerator to a degree lower than that of the denominator.
    Thus:
    (x^3)/((x^2)+1)
    =x - x/((x^2)+1)
    =x - (1/2)(2x)/(x²+1)
    the first term can be integrated using the simple power rule.
    We note that the second term has been transformed into the form where the numerator is the derivative of the denominator. The integral of the second term is thus -(1/2)ln(denominator).

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