Integrating Functions Using Long Division and Completing the Square
Use long division or completing the square to expose a basic integral without changing its domain.
College Board Topic 6.10 · CED p. 122 (PDF p. 127) · CED effective Fall 2020; project course outline 2026–27.
Sometimes algebra is the integration technique. For a rational expression with numerator degree at least denominator degree, divide first. For example, (x²+1)/(x+1)=x−1+2/(x+1), so basic power and logarithm rules do the work. State where the original denominator is nonzero.
Completing the square can reveal an inverse-trigonometric form: x²+4x+8=(x+2)²+4. With u=(x+2)/2, ∫dx/(x²+4x+8)=½arctan((x+2)/2)+C. Squaring a denominator or dropping a scale factor changes the problem. This topic needs long division and completing the square in AB; BC’s linear partial-fraction decomposition is a separate extension.
Let's work through it
- Divide (x²+1) by x+1 to get x−1 with remainder 2. Reason: The polynomial part must be removed before integrating the rational remainder.
- Integrate x−1+2/(x+1) to get x²/2−x+2ln|x+1|+C. Reason: Each term now matches a basic rule.
- Restrict to intervals excluding x=−1. Reason: The original rational function is undefined there.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Rewrite x²−6x+13 by completing the square.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Rewrite x²−6x+13 by completing the square.
Worked answer and reason
(x−3)²+4.
Add and subtract 9 to make a perfect square.
Retrieval 2
When should you divide a rational integrand first?
Worked answer and reason
When numerator degree is at least denominator degree.
Division separates a polynomial from a proper rational remainder.
Catch the tempting mistake
- Completing the square changes x²+4x+8 to (x+2)²+8.
- Expanding gives x²+4x+12. The correct leftover constant is 4.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 24 · No calculator · analytic
- Unit 6 problem 25 · No calculator · analytic
- Unit 6 problem 27 · No calculator · analytic
For the teacher
- Have students verify algebra before integrating and check the final result by differentiation. Keep BC-only partial fractions separate from AB-required algebra.
Glossary and deeper reading · 2 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Logarithm
A logarithm gives the exponent required to obtain a number from a specified base. For real logarithms, the argument is positive and the base is positive and different from 1. For example, log₂8=3 because 2³=8, while ln uses the base e. Logarithm rules convert products into sums and powers into constant multiples, subject to the domain restrictions.
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Completing the square
Completing the square rewrites a quadratic using a squared linear expression plus a constant. For a monic quadratic x²+bx+c, add and subtract (b/2)² to obtain (x+b/2)²+c−(b/2)². For example, x²+4x+7 becomes (x+2)²+3. The rewritten form can reveal a vertex or prepare an integral for substitution.