Calculus I · Unit 3A · lesson
Substitution: Reversing the Chain Rule
Learn Substitution: Reversing the Chain Rule through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Computing integralsWhat this section is building
Learn Substitution: Reversing the Chain Rule through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.
Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.
Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.
Learning objectives
Recognize chain-rule structure, choose a useful substitution, transform the differential, and return to the original variable.
Substitution: Reversing the Chain Rule
Substitution reverses the chain rule
A composite derivative has the pattern . Substitution recognizes that pattern inside an integral, names the inner expression , and replaces the accompanying derivative factor with . The purpose is not to change letters for entertainment; it is to turn a complicated composite integrand into a basic antiderivative in one variable.
A good substitution accounts for every factor, including the differential. After choosing , compute explicitly and compare it with what remains in the integrand. Constant multiples can be adjusted, but missing variable factors cannot be wished into existence. For an indefinite integral, return to the original variable at the end and verify by differentiating through the chain rule.
The chain rule says
Reversing it gives
The notation , organizes this reversal.
A complete substitution
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Worked solution
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u3a-substitution-01Evaluate .
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Let , so .
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A substitution is incomplete if both and remain in the transformed integral. Every factor must be rewritten or accounted for before integration.
Read this graph as text
Substitution as relabeling a composed differential. A flow diagram shows an inner function and its derivative being replaced by u and du. Map inner expression u=g(x), differential du=g prime dx, and simplified integral. Do not teach substitution as arbitrary symbol swapping.
Every relationship in substitution as relabeling a composed differential uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Map inner expression u=g(x), differential du=g prime dx, and simplified integral.
Substitution as relabeling a composed differential. Map inner expression u=g(x), differential du=g prime dx, and simplified integral.
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