What if the derivative is missing a constant multiplier?

Multiply and divide the integral by the required constant to match the differential $du$. Constants can be moved in and out of the integral sign.

Do I have to change the limits of integration for definite integrals?

You must either change the limits to correspond to $u$ and evaluate directly, or compute the antiderivative in terms of $x$ first and use the original limits. Both methods are valid.

What if a single u-substitution leaves extra x variables?

If $x$ remains after substitution, try expressing those remaining $x$ variables in terms of $u$ (algebraic substitution), or consider another integration technique such as integration by parts.

How to integrate by u-substitution

U-substitution reverses the chain rule to evaluate integrals of the form $\int f(g(x))g'(x) dx$ . It applies when the integrand contains a composite function multiplied by the derivative of its inner function, up to a constant factor.

The setup

Identify the inner function $u = g(x)$ . Compute its differential $du = g'(x) dx$ . Determine if $g'(x) dx$ or a constant multiple of it is present in the original integrand.

The steps

Substitute $u$ and $du$ into the integral to eliminate all instances of $x$ . 2. Evaluate the new integral with respect to $u$ . 3. For indefinite integrals, replace $u$ with the original expression $g(x)$ and add the constant of integration $C$ . For definite integrals, either convert the limits of integration to $u$ -values or revert to $x$ before evaluating.

Checking the result

Differentiate the resulting antiderivative with respect to $x$ . The derivative must match the original integrand exactly. If it does not, verify the differential $du$ and the antiderivative steps.

Common errors

Failing to substitute $dx$ completely and mixing $x$ and $u$ variables in the same integral. Forgetting to update the limits of integration when evaluating a definite integral entirely in terms of $u$ . Choosing a $u$ whose derivative does not appear as a factor in the integrand.

Worked example

Evaluate $\int x^2 e^{x^3} dx$ .

Let $u = x^3$ . Compute the differential: $du = 3x^2 dx$ . Rearrange to isolate the terms in the integral: $\frac{1}{3} du = x^2 dx$ . Substitute $u$ and $du$ into the integral: $\int e^u (\frac{1}{3} du)$ . Factor out the constant: $\frac{1}{3} \int e^u du$ . Evaluate the integral: $\frac{1}{3} e^u + C$ . Substitute back $u = x^3$ : $\frac{1}{3} e^{x^3} + C$ .

FAQ

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Answer form

References: Stewart, J. (2015). Calculus (8th ed.). Cengage Learning. · OpenStax. (2016). Calculus Volume 1. OpenStax. Chapter 5.5.