My analysis book makes the following statement:

Every rational function with real coefficients can be integrated in terms of

  • rational functions,
  • logarithm functions,
  • arctangent functions.

Does it mean that after polynomial division, partial fraction expansion, completing the square, substituting where needed, and knowing that

\begin{align} \\ \int \frac{dx}{(x-a)^n} &= -\frac{1}{n-1} \frac{1}{(x-a)^{n-1}} \ \ \ (n \neq 1) \\ \int \frac{dx}{x-a} &= \log|x-a| \\ \int \frac{dx}{x^2+1} &= \arctan x \end{align}

one can integrate any rational function?

  • 3
    $\begingroup$ In principle. In reality, you will often not be able to split the polynomials into irreducible factors. $\endgroup$ – Daniel Fischer Jun 28 '13 at 18:22
  • $\begingroup$ You also need to be able to compute $$\int \frac{dx}{(x^2+1)^n}$$ for $n > 1$. $\endgroup$ – mrf Jun 28 '13 at 18:55
  • $\begingroup$ @mrf: I saw this one in many materials, yet I didn't quite understand why should we pay special attention to the integral if it can be quickly reduced to the 1st case after substitution and partial integration? $\endgroup$ – Leo Jun 28 '13 at 19:07
  • $\begingroup$ @Leo If you know complex numbers you can do it even without arctangent... You can just reduce it to the first and second formula (always). $\endgroup$ – Kartik Aug 11 '16 at 14:56

You're in part right, as compared to this page. Yes, you need to know about polynomial division, but you also need to know about the degree of the numerator and the denominator, beacuse (and this comes from the link) the degree of the numerator and denominator determines what to do first. If the degree of the numerator is greater than that of the denominator, you do polynomial division, otherwise, you need to factor the denominator. Then you do partial fraction expansion (like you said). Yes, you may need to know the conditions you stated because the rational function could be of any of those forms.

This is the best thing I can say, but I hope this helps!


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