Integration using long division
Also known as: integrating rational functions by long division, polynomial long division in integration
Integration using long division is a technique for integrating a rational function whose numerator degree is greater than or equal to its denominator degree. Dividing first rewrites the integrand as a polynomial plus a simple remainder term that can be integrated directly.
Some rational functions cannot be integrated in the form they are given. When the degree of the numerator is greater than or equal to the degree of the denominator, the fraction is called improper, and no basic antiderivative rule applies. Polynomial long division fixes this: dividing the numerator by the denominator produces a polynomial quotient plus a proper remainder fraction, and each piece integrates with rules you already know.
Consider ∫ (x² + 3x + 5) / (x + 1) dx. Long division gives x + 2 with a remainder of 3, so the integrand becomes x + 2 + 3/(x + 1). Integrating term by term yields x²/2 + 2x + 3ln|x + 1| + C. The remainder term produced a natural logarithm, which is the typical outcome when the leftover denominator is linear.
A quick test tells you when to divide. If the numerator's degree is greater than or equal to the denominator's degree, divide first. If the numerator's degree is already smaller, long division is not needed — instead look at whether the denominator factors (partial fractions), whether the numerator is the derivative of the denominator (u-substitution), or whether the denominator is an irreducible quadratic (complete the square and expect an inverse tangent). Recognizing which of these situations you are in is most of the work.
On the AP Calculus AB exam, this technique appears in the integration unit and is usually paired with completing the square as the two setup moves for rational integrands. Free-response and multiple-choice questions rarely announce the method — they simply present an improper rational function, and you are expected to notice the degree comparison and divide before integrating.
Key takeaways
- Use long division when the numerator's degree is greater than or equal to the denominator's degree.
- Division rewrites the integrand as a polynomial plus a proper remainder fraction that integrates easily.
- A linear leftover denominator typically produces a natural logarithm term, such as 3ln|x + 1|.
- If the numerator's degree is already lower, use u-substitution, partial fractions, or completing the square instead.
- AP Calculus AB tests the technique without naming it — you must spot the improper rational function yourself.