Code Scholars Tutoring · AP Calculus AB

Algebra before integration and method selection

Unit 6 · Topics 6.10, 6.14. Original AP Calculus AB notes, worked example, and self-check.

Understand the idea

Before selecting a technique, simplify the integrand: divide a higher-degree polynomial by its denominator or complete a quadratic square. Match known derivative patterns, especially logarithms and inverse tangent. AB integration methods include substitution and algebraic rewriting; integration by parts, partial-fraction decomposition, and improper integrals belong to BC. A difficult-looking integrand may not have an elementary AB antiderivative, so a numerical definite integral may be appropriate.

\int\frac{dx}{a^2+x^2}=\frac1a\arctan\left(\frac xa\right)+C\quad(a>0)

Worked example

Find $\displaystyle \int x^2/(x+1)\,dx$ .

Divide first:

\displaystyle x^2/(x+1)=x-1+1/(x+1)

. Integrate termwise to get

\displaystyle x^2/2-x+\ln|x+1|+C

, on an interval not containing

\displaystyle -1

Try it yourself

Evaluate $\displaystyle \int dx/(x^2+2x+2)$ .

Show solution and reasoning

Complete the square: $\displaystyle x^2+2x+2=(x+1)^2+1$ . The antiderivative is $\displaystyle \arctan(x+1)+C$ .

Unit 6 MCQ