Integration with long division and completing the square — AP Calculus AB
1. Integration of Improper Rational Functions via Long Division ★★☆☆☆ ⏱ 4 min
For a rational function $f(x) = \frac{N(x)}{D(x)}$, if the degree of the numerator $N(x)$ is greater than or equal to the degree of the denominator $D(x)$, the function is called improper. Improper rationals cannot be integrated directly with basic rules, so we use polynomial long division to rewrite them into a form we can integrate.
\frac{N(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}, \quad \deg(R(x)) < \deg(D(x))
Where $Q(x)$ is the quotient polynomial and $R(x)$ is the remainder. By linearity of integration, we can integrate each term separately using rules you already know.
2. Completing the Square for Irreducible Quadratics ★★★☆☆ ⏱ 5 min
When you have a proper rational function with a quadratic denominator, first calculate the discriminant $b^2 - 4ac$. If the discriminant is negative, the quadratic is irreducible (cannot be factored over the reals), so we use completing the square to rewrite it to match the inverse tangent antiderivative form.
- Factor the leading coefficient $a$ out of the first two terms: $a\left(x^2 + \frac{b}{a}x\right) + c$
- Add and subtract $\left(\frac{b}{2a}\right)^2$ inside the parentheses: $a\left[\left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\right] + c$
- Simplify the constant term: $a(x+h)^2 + k$, where $h = \frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$
\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C
3. Evaluating Definite Integrals ★★★☆☆ ⏱ 3 min
Most AP exam questions on this topic ask for a definite integral, which combines the algebraic preprocessing above with the Fundamental Theorem of Calculus, Part 2 (FTC 2): $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$. Always confirm the integrand is continuous over the entire interval of integration.
4. AP-Style Concept Check ★★★★☆ ⏱ 2 min
Common Pitfalls
Why: Students rush into integration without checking the form, assuming all rational functions work with u-substitution
Why: Students forget to multiply the subtracted constant by the leading coefficient when moving it outside parentheses
Why: Students do not match the numerator to the derivative of the denominator, leading to an unsolvable integral
Why: Students mix up derivative and integral rules for inverse tangent
Why: Students rush on exam day and misremember the order of subtraction