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Calculus I · Unit 3A · lesson

Substitution: Reversing the Chain Rule

Concept

Learning objectives

Recognize chain-rule structure, choose a useful substitution, transform the differential, and return to the original variable.

Substitution: Reversing the Chain Rule

Explanation

Substitution reverses the chain rule

A composite derivative has the pattern F′(g(x))g′(x)F'(g(x))g'(x). Substitution recognizes that pattern inside an integral, names the inner expression u=g(x)u=g(x), and replaces the accompanying derivative factor with dudu. 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 uu, compute dudu 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

ddxF(g(x))=F′(g(x))g′(x).\frac{d}{dx}F(g(x))=F'(g(x))g'(x).

Reversing it gives

∫F′(g(x))g′(x) dx=F(g(x))+C.\int F'(g(x))g'(x)\,dx=F(g(x))+C.

The notation u=g(x)u=g(x), du=g′(x) dxdu=g'(x)\,dx organizes this reversal.

Guided walkthrough

A complete substitution

Evaluate

∫2x(x2+5)7 dx.\int 2x(x^2+5)^7\,dx.
Answer reveal

Worked solution

Write a real attempt before opening the supplied answer.

Interactive checku3a-substitution-01

Evaluate ∫(3x+1)4 dx\int(3x+1)^4\,dx.

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Show hint

Let u=3x+1u=3x+1, so du=3dxdu=3dx.

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Common mistake

A substitution is incomplete if both xx and uu remain in the transformed integral. Every factor must be rewritten or accounted for before integration.

Substitution as relabeling a composed differential. Map inner expression u=g(x), differential du=g prime dx, and simplified integral.
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.

Visual study

Substitution as relabeling a composed differential. Map inner expression u=g(x), differential du=g prime dx, and simplified integral.

After the explanation

Use the section idea

Reading lens

Choose an integration method from the integrand's structure, then verify the result by differentiation.

Mental model

Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.

Decision

Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.

Common trap

Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.

Check yourself

Can you name the derivative rule being reversed and differentiate the final answer back to the integrand?

Source & rights

Original instruction with traceable references.

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