Substitution Rule

The central idea

The substitution rule reverses the chain rule. If $g$ is continuously differentiable and $f$ is continuous on the relevant range (or $f$ has an antiderivative there), then $\int f(g(x))g'(x)\,dx=\int f(u)\,du$ with $u=g(x)$. For definite integrals, either change the bounds to $u=g(a),g(b)$ and stay in $u$, or return to $x$ before using the original bounds.

Definitions, hypotheses, and notation

A substitution is complete only when the transformed integral contains no unintended original variable. Sometimes algebra must express a leftover factor in terms of $u$; sometimes that difficulty signals a poor choice. Constant multiples are harmless because they may be factored in or out, but a genuinely unmatched factor cannot simply be discarded.

Substitution may reverse definite-integral bounds. If $u=g(x)$ decreases on the interval, then $g(a)>g(b)$, and the reversed $u$-bounds automatically encode orientation. Do not reorder them without inserting a minus sign. Differentiating the final antiderivative or numerically checking the definite value confirms both the derivative factor and the orientation.

Conceptual meaning

A good substitution treats a repeated inner expression as one quantity and absorbs its differential. It changes the coordinate used to measure accumulation; the derivative factor accounts for stretching or reversing that coordinate.

A dependable method and decision rule

1. Look for a composite expression and a constant multiple of its derivative.
2. Set $u$ equal to the inner expression and compute $du$.
3. Rewrite every factor, including $dx$, in terms of $u$.
4. Integrate with respect to $u$.
5. For an indefinite integral substitute back; for a definite one use one consistent bound strategy.

Fully worked example

Graphical or geometric meaning

As $x$ runs from $0$ to $1$, $u=x^2+1$ runs monotonically from $1$ to $2$. The factor $2x$ is exactly the rate at which the $u$-coordinate changes, so $2x dx$ becomes one unit of differential $du$.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate an indefinite answer using the chain rule.
• Verify transformed bounds by direct substitution into $u=g(x)$.
• Estimate sign and magnitude of the original definite integral.

State why the change of variable is legal

A standard definite version assumes $g$ is continuously differentiable and $f$ is continuous on the range traced by $g$, giving $\int_a^b f(g(x))g'(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du$. An antiderivative hypothesis supports the same chain-rule argument. For an indefinite integral, transform every factor including $dx$ and then return to the original variable. For a definite integral, either change the bounds and stay in $u$, or return to $x$ before using the original bounds. Mixing those approaches is a common error. Differentiate the final antiderivative to verify the inner derivative and every constant factor.

Practice

1. Evaluate $\int 3x^2e^{x^3}dx$.
2. Evaluate $\int_0^2 x(x^2+1)dx$.
3. For $u=5x-2$, what is $dx$?

Connections and next steps

• Chain Rule
• Antiderivatives
• Definite Integral