Code Scholars Tutoring · AP Calculus BC

Algebraic rewriting before integration

Unit 6 · Topics 6.10 · Detailed notes, a worked example, and a self-check with reasoning.

Browse Unit 6 lessons

Understand the idea

Algebra can expose an antiderivative before a more elaborate technique is needed. Compare polynomial degrees and divide when the numerator degree is at least the denominator degree. Complete a quadratic square when it reveals an inverse-tangent pattern. Keep track of the original domain while rewriting the expression. In BC, this algebraic preparation also helps decide whether substitution, integration by parts, or linear partial fractions is the most useful next step.

\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 $\displaystyle \arctan(x+1)+C$ .