AP Calculus AB · Unit 6: Integration and Accumulation of Change
6.10 Integrating Functions Using Long Division and Completing the Square
In short
You must be able to integrate functions by first rearranging the integrand into an equivalent form using algebraic techniques, specifically long division and completing the square. This applies to both indefinite and definite integrals.
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What else do you need to know about Integrating Functions Using Long Division and Completing the Square?
You should be able to recognize when a rational integrand requires long division (e.g., when the degree of the numerator is greater than or equal to the degree of the denominator) or completing the square (e.g., for quadratic denominators that do not factor easily) to match standard integration formulas. The focus is on selecting the appropriate strategy based on the integrand's form, recognizing opportunities to apply mathematical rules to simplify integration.
Can you do these?
- I can determine indefinite integrals and evaluate definite integrals for integrands requiring rearrangements into equivalent forms.
- I can use techniques such as long division and completing the square to rearrange integrands into equivalent forms to find antiderivatives.
How is Integrating Functions Using Long Division and Completing the Square tested on the exam?
When evaluating an integral without a calculator, you must present an appropriate antiderivative and include a constant of integration (+ C) for indefinite integrals. For definite integrals evaluated with a calculator in free-response questions, you must present the full definite integral expression, including endpoints and the differential (e.g., dx). You must be careful with parentheses usage and avoid 'linking' unequal expressions with strings of equal signs. A common pitfall is failing to recognize the correct algebraic manipulation required before integrating.
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Integrating Functions Using Long Division and Completing the Square: common questions
What do you need to know about Integrating Functions Using Long Division and Completing the Square for AP Calculus AB?
You must be able to integrate functions by first rearranging the integrand into an equivalent form using algebraic techniques, specifically long division and completing the square. This applies to both indefinite and definite integrals.
How is Integrating Functions Using Long Division and Completing the Square tested on the AP Calculus AB exam?
When evaluating an integral without a calculator, you must present an appropriate antiderivative and include a constant of integration (+ C) for indefinite integrals. For definite integrals evaluated with a calculator in free-response questions, you must present the full definite integral expression, including endpoints and the differential (e.g., dx).
How do I know I have mastered Integrating Functions Using Long Division and Completing the Square?
Check you can do each of these: determine indefinite integrals and evaluate definite integrals for integrands requiring rearrangements into equivalent forms; use techniques such as long division and completing the square to rearrange integrands into equivalent forms to find antiderivatives.
Where does Integrating Functions Using Long Division and Completing the Square fit in AP Calculus AB?
Integrating Functions Using Long Division and Completing the Square is topic 6.10 in Unit 6: Integration and Accumulation of Change of AP Calculus AB.
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