Integration with long division and completing the square — AP Calculus BC
1. Integration of Improper Rational Functions via Polynomial Long Division ★★☆☆☆ ⏱ 4 min
A rational function $\frac{P(x)}{Q(x)}$ is improper if $\deg(P(x)) \geq \deg(Q(x))$, and cannot be integrated directly using standard templates. Polynomial long division rewrites the improper integrand as:
\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}
where $S(x)$ is the quotient polynomial, and $R(x)$ is the remainder polynomial with $\deg(R(x)) < \deg(Q(x))$. This is analogous to rewriting an improper fraction like $\frac{15}{4}$ as $3 + \frac{3}{4}$: we separate the whole polynomial part from the proper rational remainder, which can then be integrated with other techniques. The entire expression is integrated term-by-term: $S(x)$ uses the power rule, and $\frac{R(x)}{Q(x)}$ is handled with other methods.
Exam tip: Always check the degree of the numerator and denominator before starting integration. If $\deg(P) \geq \deg(Q)$, you must do long division first — skipping this step guarantees an incorrect result.
2. Completing the Square for Irreducible Quadratic Denominators ★★★☆☆ ⏱ 4 min
After long division (or when starting with a proper rational function), you may encounter an irreducible quadratic denominator $ax^2 + bx + c$, where the discriminant $b^2 - 4ac < 0$, so it cannot be factored into real linear terms. Completing the square rewrites the quadratic to match one of two standard inverse trigonometric integral templates.
To complete the square for $ax^2 + bx + c$: 1. Factor the leading coefficient $a$ out of the first two terms. 2. Add and subtract $\left(\frac{b}{2a}\right)^2$ inside the parentheses to form a perfect square, resulting in $a u^2 + k$ where $u = x + \frac{b}{2a}$. This matches the form needed for substitution into the standard templates:
\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C
\int \frac{1}{\sqrt{a^2 - u^2}} du = \arcsin\left(\frac{u}{a}\right) + C
Exam tip: Always factor out the leading coefficient of the quadratic before completing the square if it is not 1. Forgetting this step gives an incorrect constant term that will not match standard inverse trig forms.
3. Integrating Linear Over Irreducible Quadratic Functions ★★★★☆ ⏱ 4 min
A common AP exam integrand is $\frac{mx + n}{ax^2 + bx + c}$ where the denominator is irreducible. You cannot directly apply inverse trig rules here; instead split the numerator into a multiple of the derivative of the denominator plus a constant. The method is:
- Let $D(x) = ax^2 + bx + c$, compute $D'(x) = 2ax + b$
- Write $mx + n = A(2ax + b) + B$, solve for constants $A$ and $B$ by equating coefficients
- Split the integrand into two terms: $\frac{A D'(x)}{D(x)} + \frac{B}{D(x)}$
- Integrate the first term with the logarithmic rule, and the second term by completing the square and inverse trig rule
The first term is always of the form $\frac{f'(x)}{f(x)}$, which integrates to $\ln|f(x)|$. Since irreducible quadratics are always positive, you can drop the absolute value bar for simplicity.
Exam tip: Always drop the absolute value around an irreducible quadratic in the log term, since the quadratic is always positive and never crosses the x-axis, simplifying your final answer.
4. AP-Style Concept Check ★★★☆☆ ⏱ 2 min
Common Pitfalls
Why: Students focus on completing the long division calculation and omit the non-zero remainder, which is present in most problems.
Why: Students think you only need long division when the numerator degree is strictly higher, not equal.
Why: The student forgot to factor out the leading coefficient 2 from the x terms before completing the square.
Why: Students see a quadratic denominator and immediately complete the square, forgetting the linear numerator produces a logarithmic term.
Why: The student forgot the substitution step for $u = 2x + 3$, which requires accounting for the chain rule factor of 2 from $du = 2 dx$.