Selecting Techniques for Antidifferentiation
Select an integration method from the integrand and inspect the interval before exact or numerical evaluation.
College Board Topic 6.14 · CED p. 126 (PDF p. 131) · CED effective Fall 2020; project course outline 2026–27.
Pause before integrating. Is it a basic derivative in reverse? Is there an inside function with its derivative nearby? Can factoring, division, or completing the square expose a basic form? For AB, these are the main choices. Selecting a method is a skill, not a race to write symbols.
A useful decision path is: simplify algebra; recognize a basic rule; test a substitution; then decide whether the task instead calls for geometry, an integral-defined answer, or numerical approximation. A definite integral may be computable even when an indefinite integral has no elementary form. For calculator practice, first write the correct expression and interval, then enter it, report units if applicable, and round only at the end.
Let's work through it
- For ∫(2x)/(x²+4)dx, choose u=x²+4. Reason: The numerator matches the derivative of the denominator.
- For ∫(x²+1)/(x+1)dx, choose long division first. Reason: The rational function is improper and simplifies into basic pieces.
- For ∫₀¹ e^(−x²)dx, keep integral notation or use numerical integration when permitted. Reason: There is no elementary antiderivative to obtain by these AB methods.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Which method fits ∫cos(5x)dx?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Which method fits ∫cos(5x)dx?
Worked answer and reason
A linear substitution, or a reverse-chain-rule factor of 1/5.
The derivative of 5x is 5.
Retrieval 2
Before pressing the integration key, what must you specify?
Worked answer and reason
The integrand, variable, bounds, and any domain or context restrictions.
A numerical value is only meaningful for a well-defined integral.
Catch the tempting mistake
- One method should work on every integral.
- Different structures call for different methods; some integrals are best handled numerically or left in exact integral form.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 6 problem 26 · No calculator · verbal
- Unit 6 problem 27 · No calculator · analytic
- Unit 6 problem 28 · Graphing calculator required · analytic
For the teacher
- Use a card sort in which students name a method and its evidence before solving. Mix noncalculator exact work with a calculator setup item.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Integral
Integration can mean accumulating a function over an interval or finding an antiderivative. A definite integral gives a number, while an indefinite integral represents a family of antiderivatives. For example, integrating 2x indefinitely gives x²+C, and integrating it from 0 to 2 gives 4. The Fundamental Theorem of Calculus connects these two operations under its hypotheses.
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Definite integral
A definite integral accumulates a function over a specified interval as a limit of Riemann sums when that limit exists. Its sign includes the function’s sign and the orientation of the limits. For example, the integral of the constant −2 from 0 to 3 is −6, although the geometric area is 6. In a rate model, the result has rate units multiplied by input units.