u-Substitution — Mathematics Knowledge System

Overview

u-Substitution (the substitution rule) is the integral counterpart of the chain rule. It simplifies an integral by replacing a complicated expression with a single variable u, transforming the integral into a more recognizable form in terms of u.

Intuition

If an integrand looks like f(g(x))·g'(x), it's the output of the chain rule run in reverse. Substituting u=g(x) collapses g'(x)dx into du, turning a messy composite-function integral into a simple ∫f(u)du.

Interactive Graph

Formal Definition

Definition

If u = g(x) is differentiable and f is continuous on the range of g:

\int f(g(x))\,g'(x)\,dx = \int f(u)\,du \quad \text{where } u=g(x),\ du=g'(x)\,dx

u-Substitution (indefinite integral)

\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du

The limits of integration must also be converted to u-values

u-Substitution (definite integral)

Applications

Computing work done by a variable force often requires substitution to simplify the force function before integrating over displacement.

Worked Examples

Let u = x²+1, so du = 2x dx — which appears exactly in the integrand.
$u=x^2+1,\ du=2x\,dx$
Substitute and integrate using the power rule.
$\int u^5\,du = \frac{u^6}{6} + C$
Substitute back u=x²+1.
$= \frac{(x^2+1)^6}{6} + C$

Answer: (x²+1)⁶/6 + C

Practice Problems

Difficulty 3/10

Evaluate ∫ cos(3x) dx.

Difficulty 5/10

Evaluate ∫ (ln x)²/x dx.

Common Mistakes

Common Mistake

Forgetting to convert the limits of integration to u-values when substituting in a definite integral.

Either convert the limits to g(a) and g(b) and evaluate entirely in u, or substitute back to x before plugging in the original limits — mixing the two approaches gives a wrong answer.

Common Mistake

Choosing a substitution u=g(x) when g'(x) doesn't appear (up to a constant) anywhere in the integrand.

u-substitution only works cleanly when the integrand contains both g(x) and a factor proportional to g'(x); otherwise a leftover x-term will remain that can't be expressed in terms of u.

Quiz

u-substitution is the integral counterpart of which differentiation rule?

the product rulethe chain rulethe quotient ruleL'Hôpital's rule

When applying u-substitution to a DEFINITE integral, you should:

keep the original x-limitsconvert the limits to the new variable u (or back-substitute before evaluating)drop the limitsdouble the limits

Summary

u-Substitution reverses the chain rule: ∫f(g(x))g'(x)dx = ∫f(u)du where u=g(x).
Look for an inner function g(x) whose derivative g'(x) also appears in the integrand.
For definite integrals, either convert the bounds to u-values or substitute back to x before evaluating.
Always add the constant of integration C for indefinite integrals.

References

WebsiteWikipedia — Integration by substitution