Math101 · Independent Ontario learning library
Math101Substitution Rule
A rigorous, example-driven guide to substitution rule, including hypotheses, method choice, verification, and practice.
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
- Look for a composite expression and a constant multiple of its derivative.
- Set $u$ equal to the inner expression and compute $du$.
- Rewrite every factor, including $dx$, in terms of $u$.
- Integrate with respect to $u$.
- For an indefinite integral substitute back; for a definite one use one consistent bound strategy.
