Integrating Functions Using Long Division / Selecting Techniques for Antidifferentiation 
(Topics 6.10 and 6.14)

Unit 6 - Day 16

​Learning Objectives​
  • Use equivalent forms of integrands to evaluate integrals

  • Select appropriate techniques for antidifferentiation

​Success Criteria
  • I can re-write 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
Activity: Integration Techniques Card Sort 



Lesson Handout


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 warm-up 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 completing-the-square 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 u-Substitution.  To check their work quickly, each set of cards spells a word. 

  • Basic Antidifferentiation Rules: PLAY

  • Simplifying: ROUGH

  • U-sub: 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!

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