Integration by Substitution (u-substitution)

Integration by substitution, often called u-substitution, is a calculus method used to solve complex integrals by replacing a part of the integrand with a single variable u. Substitution is a technique that simplifies the integration of functions that are the result of a chain-rule derivative. The term ‘substitution’ refers to changing variables or substituting the variable u and du for appropriate expressions in the integrand. This simplifies the expression so it matches a standard integral formula.

Contents

U-substitution is a powerful technique for finding antiderivatives, especially when dealing with functions that are products or compositions of other functions. The core idea is to simplify the integral by changing the variable from x to u, where u is typically a part of the original function that, when differentiated, helps simplify the rest of the integral. This method is applicable for both indefinite and definite integrals.

Substitution Rule for Indefinite Integrals

If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

$\int f(g(x)) g^{\prime}(x) dx=\int f(u) du, \text { where } u=g(x)$

Substitution Rule for Definite Integrals

If g′ is continuous on [a,b] and f is continuous on the range of u=g(x), then

$\int_a^b{f\left( g\left( x \right) \right) g^{\prime}\left( x \right) \,\,}=\,\,\int_{g\left( a \right)}^{g\left( b \right)}{f\left( u \right) du}$

Calculus Note

When using the Substitution Rule for integrating definite integrals, it is important to change the limits of integration from those of the original function to those of the substituted function. Otherwise, the definite integral will evaluate to an incorrect result.

The Basic Process of U-Substitution

The method of U-substitution involves four key steps:

Identify u and du: Choose an expression within the integrand to be u. Then, find the derivative of u with respect to x, which gives you du.
Solve for dx: Rearrange the du equation to express dx in terms of du and x.
Substitute into the integral: Replace u and dx in the original integral. The goal is to eliminate all x variables, leaving an integral solely in terms of u and du. If x variables remain, you might need to express x in terms of u from your initial u definition.
Integrate and back-substitute: Find the antiderivative with respect to u, and then replace u with its original expression in terms of x. Don’t forget the constant of integration, C, for indefinite integrals.

Calculus Note

When to do u-substitution and when to integrate by parts?

A u-substitution can be done whenever you have something containing a function (we’ll call this g), and that something is multiplied by the derivative of g.

Integration by parts is whenever you have two functions multiplied together–one that you can integrate, one that you can differentiate.

My strategy is to try to “play it out” in my mind and try to see which one will work better. The best way to get better at these sorts of integrals is to practice large sets of each type. Then, you start to think “Oh–this looks like a u-sub!” or, “maybe by-parts is better for this.” Practice is really the best way to get better at recognizing each type.

apnorton. “When to do u-substitution and when to integrate by parts.” Mathematics Stack Exchange, October 24, 2013. https://math.stackexchange.com/q/538663.

Practice Problems

Use the following websites to practice trigonometric substitution.

“5.5E and 5.6E u-Substitution Exercises.” LibreTexts, December 29, 2018. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_211_Calculus_II/Chapter_5%3A_Integration/5.5E_and_5.6E_u-Substitution_Exercises.

Dawkins, Paul. “Calculus I.” Substitution Rule for Indefinite Integrals (Practice Problems). Paul’s Online Notes, May 5, 2024. https://tutorial.math.lamar.edu/problems/calci/substitutionruleindefinite.aspx.

Foster, Joe. “u-Substitution.” University of South Carolina, Accessed July 16, 2026. https://people.math.sc.edu/josephcf/Teaching/142/Files/Worksheets/U-Substitution.pdf

“Integration by Substitution Practice Problems.” GeeksforGeeks, August 1, 2024. https://www.geeksforgeeks.org/maths/integration-by-substitution-practice-problems/.

“Substitution Rule.” SFU, Accessed July 16, 2026. https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_SubRule.html.

References

“Substitution Rule.” SFU, Accessed July 16, 2026. https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_SubRule.html.

[ ℰ ] “U-Substitution Integration Study Guide – Mathematics.” Studley, Accessed July 16, 2026. https://www.studley.ai/study-sets/mathematics/u-substitution-integration-method.

Additional Reading

Dawkins, Paul. “Calculus I.” Substitution Rule for Indefinite Integrals, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/substitutionruleindefinite.aspx.

“Integration by Substitution.” Math Is Fun, Accessed July 16, 2026. https://www.mathsisfun.com/calculus/integration-by-substitution.html.

“Integration by Substitution.” Wikipedia, May 10, 2019. https://en.wikiversity.org/wiki/Integration_by_Substitution.

“Integration by Substitution.” Wikipedia, June 19, 2026. https://en.wikipedia.org/wiki/Integration_by_substitution.

Khandelwal, Neha. “Integration by Substitution.” CK-12 Foundation, July 10, 2026. https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-12/section/7.3/primary/lesson/integration-by-substitution/.

OpenStax. “5.5: U-Substitution.” Libretexts, December 4, 2018. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_210_Calculus_I_(Professor_Dean)/Chapter_5%3A_Integration/5.5%3A_U-Substitution.

[ ℰ ] McLean, Rachel. “What Is U-Substitution?” Outlier, January 29, 2022. https://articles.outlier.org/what-is-u-substitution.

Videos

Calculus 1: The Substitution Rule (Section 5.5) | Math with Professor V

Examples of using the substitution rule (u-substitution) to evaluate indefinite and definite integrals. Review of even and odd functions and using symmetry to evaluate antiderivatives.

Integration Using U-Substitution | Calculus 1 | Math with Professor V

Examples of using the substitution rule (u-substitution) to evaluate indefinite and definite integrals. Lots of examples to help you understand how to choose u appropriately to evaluate an integral.

How To Integrate Using U-Substitution

This calculus video tutorial provides a basic introduction into u-substitution. It explains how to integrate using u-substitution. You need to determine which part of the function to set equal to the u variable and you to find the derivative of u to get du and solve for dx. After replacing all x variables with u variables, find the antiderivative of f(u) and substitute u in the new function with x variables. This video contains plenty of examples and practice problems of finding the indefinite integral using u-substitution. Examples include polynomial functions, trigonometric functions, exponential functions, square root functions, and rational functions.

[ ℰ ] This exceptional reference is highly recommended for your consideration.

The featured image on this page is from the Integral Wallpaper website.