Selecting Techniques for Antidifferentiation
Combine the shared AB method-selection lesson with its genuine BC addition: parts, distinct linear partial fractions, and interval limits.
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.
Take it further in BC
BC extension to the shared AB technique-selection lesson: after simplifying, recognizing a basic rule, or trying substitution, ask whether a product suggests parts or distinct linear factors suggest partial fractions. Separately inspect the interval for a pole or infinite endpoint. “Improper” describes how an integral is defined; you may still use substitution, parts, or partial fractions inside each finite piece.
For x eˣ use parts; for 1/[x(x+1)] use partial fractions; for ∫₁^∞ 1/[x(x+1)]dx use partial fractions and then an endpoint limit. Choose the method because of structure and verify by differentiation or limiting behavior. A calculator approximation can support a finite, continuous definite integral, but it cannot establish ordinary convergence merely by returning a number.
- For ∫₁^∞ 1/[x(x+1)]dx, choose partial fractions: 1/x−1/(x+1). Reason: Distinct linear factors expose basic logarithms.
- On [1,b], the value is ln(b/(b+1))+ln 2. Reason: Integrate the proper rational function on a finite continuous interval.
- Take b→∞ to obtain ln 2. Reason: The ratio tends to 1; the finite limit proves convergence.
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.
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 comes first for x³/(x²−1)?
Worked answer and reason
Long division, then partial fractions for the proper remainder.
Simplifying first exposes the applicable structure.
Retrieval 2
Does a numerical answer for a singular integral prove convergence?
Worked answer and reason
No.
Convergence requires the appropriate finite one-sided limits.
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.
- BC methods replace the AB methods.
- They expand the toolbox. Basic rules, algebra, substitution, geometry, and numerical reasoning remain part of the full BC course.
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 19 · Graphing calculator required · analytic
- Unit 6 problem 30 · No calculator · 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.
- Mix AB and BC structures in a method-selection discussion. Make students identify both technique and any needed limiting process.
Glossary and deeper reading · 7 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.
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Partial fractions
Partial fraction decomposition writes a rational expression as a sum of simpler fractions. If the numerator’s degree is at least the denominator’s, polynomial division comes first. For example, 1/[x(x+1)] equals 1/x−1/(x+1) wherever the original expression is defined. The factor types and any repeated factors determine the form of the decomposition.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.
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Limit
A limit describes the value approached by outputs as inputs approach a specified target or grow without bound. For an input approaching a finite target, its value at that exact input does not determine the limit. For example, (x²−1)/(x−1) approaches 2 as x approaches 1 even though the original expression is undefined at 1. A finite two-sided limit requires the left and right approaches to agree.