Concept preview

What you will be able to do

1. 01Identify an improper rational integrand (numerator degree $\ge$ denominator degree) and rewrite it by polynomial long division as a polynomial quotient plus a proper remainder term before integrating.
2. 02Complete the square in a quadratic denominator to convert an integral into a recognisable arctangent form $\displaystyle\int\dfrac{du}{u^2+a^2}=\dfrac1a\arctan\dfrac{u}{a}+C$ (or a logarithm form when the numerator supplies the derivative of the denominator).
3. 03Combine both techniques with substitution to evaluate definite and indefinite integrals of rational functions, and avoid the two signature errors: integrating an improper fraction without dividing, and sign mistakes while completing the square.

A rational function $\dfrac{P(x)}{Q(x)}$ is called IMPROPER when the degree of the numerator is greater than or equal to the degree of the denominator, and PROPER when the numerator degree is strictly smaller. You cannot integrate an improper rational function directly — there is no antiderivative rule for, say, $\dfrac{x^2}{x^2+1}$ as it stands. The fix is polynomial long division, exactly the procedure you learned in algebra: divide $P$ by $Q$ to obtain a polynomial quotient plus a proper remainder, $\dfrac{P(x)}{Q(x)}=(\text{quotient})+\dfrac{(\text{remainder})}{Q(x)}$. The quotient is a polynomial you integrate term by term, and the leftover proper fraction is one you can handle with substitution or the arctangent form.