Integrating Using Substitution
Match an inside expression and its differential, changing both bounds or returning to the original variable.
College Board Topic 6.9 · CED p. 121 (PDF p. 126) · CED effective Fall 2020; project course outline 2026–27.
Substitution reverses the chain rule. Find an inside expression u=g(x) whose derivative g′(x)dx appears as a factor. Rewrite the entire integral in u, integrate, then return to x for an indefinite integral. Any missing constant factor must be corrected explicitly; a missing variable factor cannot simply be wished into existence.
For a definite integral, either change both endpoints to u-values and stay in u, or return to x before applying the original x-bounds. Do not combine a u-antiderivative with x-bounds. For ∫₀¹ 2x e^(x²)dx, take u=x² and du=2x dx; the bounds become 0 and 1, giving e−1. Substitution is justified by the chain rule and the Fundamental Theorem under the usual continuity conditions.
Let's work through it
- In ∫₀² x/(x²+1)dx, set u=x²+1 and du=2x dx. Reason: The denominator’s derivative matches twice the numerator.
- Use u-bounds 1 and 5: ½∫₁⁵ 1/u du. Reason: The factor ½ corrects du, and both endpoints change.
- The value is ½ln 5. Reason: Evaluate the logarithmic antiderivative with the new bounds.
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
For u=3x+1, what is 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
For u=3x+1, what is dx?
Worked answer and reason
du/3.
Differentiate u before rewriting the measure.
Retrieval 2
Can ∫e^(x²)dx use u=x² as a simple reverse-chain-rule shortcut?
Worked answer and reason
No.
The needed factor 2x is absent; changing variables produces a nontrivial remaining factor.
Catch the tempting mistake
- Substitution only changes the integrand, not dx or bounds.
- Replace the differential and, for a definite integral evaluated in u, both limits too.
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
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Put it to work
- Unit 6 problem 15 · No calculator · analytic
- Unit 6 problem 16 · No calculator · analytic
For the teacher
- Use near-miss pairs, such as 2x e^(x²) and e^(x²), to practice recognizing an actual chain-rule structure.
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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Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
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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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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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Antiderivative
An antiderivative of a function is another function whose derivative equals the original function on an interval. Two antiderivatives on the same interval differ by a constant. For example, x² is an antiderivative of 2x, and x²+7 is another. An antiderivative is a function, whereas a definite integral is an accumulated value.