Integrating Functions Using Long Division / Selecting Techniques for Antidifferentiation
(Topics 6.10 and 6.14)
Unit 6  Day 16
Unit 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Day 10
Day 11
Day 12
Day 1314
Day 15
Day 16
Day 17
Day 18
Day 19
Day 20
Day 21
All Units
Learning Objectives

Use equivalent forms of integrands to evaluate integrals

Select appropriate techniques for antidifferentiation
Success Criteria

I can rewrite the integrand in an equivalent form (completing the square, long division, etc.), when necessary, to find an antiderivative

I can select an appropriate technique for evaluating a definite integral
Quick Lesson Plan
Overview
Our card sort provides practice with several integration techniques and places the emphasis on efficiency! Finding the most efficient integration technique for solving definite and indefinite integrals is the springboard for thoughtful and productive conversation within student groups. After choosing the best method for antidifferentiation, students work collaboratively to evaluate or solve each integral.
Teaching Tips
Prior to the Topic 6.9 Card Sort, as a warmup activity, students were presented with an integral requiring long division to simplify the integrand. Rewriting integrands into a simpler form is an important option and can be achieved in many ways: using the distributive property, rewriting trig functions, factoring the numerator of a rational integrand, rewriting a rational integrand as one or more simpler fractions, or using long division prior to antidifferentiation. We will visit completingthesquare techniques during our course review, closer to the actual AP Test date.
Categories used for today’s card sort include Geometry, Basic Antidifferentiation Rules, Simplifying the Integrand, and uSubstitution. To check their work quickly, each set of cards spells a word.

Basic Antidifferentiation Rules: PLAY

Simplifying: ROUGH

Usub: STICK

Geometry: BEND
A few of the integrals can be solved with multiple strategies, so encourage students to justify their choice of solving technique.
Exam Insights
Many of the integration techniques from Unit 6 appear every year in one form or another on the AP Test. Building fluency and efficiency in antidifferentiation will save precious time and allow students to be thoughtful on more complex problems.
Student Misconceptions
Students were eager to place many integrals under the category of Basic Rules of Integration (card S, for example). A constant numerator led a few groups to a natural log antiderivative when the solution should have included the arctan function (card P). Use of the distributive property eluded many groups (card G) as did recognizing the equation for a semicircle (card B). The successful calculus student must be fluent with many mathematical processes. This card sort will cultivate that skill!