---
title: "Integration by Substitution"
description: "Reverse the chain rule by renaming an inner function, changing the differential and, for a definite integral, changing the limits with it."
canonical: https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7
source: https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Integration by Substitution

Reverse the chain rule by renaming an inner function, changing the differential and, for a definite integral, changing the limits with it.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/integration-by-substitution-mt_jH-j_XsdC7

## Ready when they can

- Choose u so that du absorbs the leftover factor in the integrand
- Convert the limits of a definite integral when substituting
- Say why a substitution fails when the required factor is missing

## Lesson: Rename the inside to undo the chain rule

Some integrands come from the chain rule. You can see an inner expression g(x) with its derivative sitting beside it. Call the inner part u, so du = g'(x) dx. Then rewrite everything in u and integrate there.

**Example.** Take the integral of 2x cos(x^2) dx. The inner expression is x^2, so set u = x^2 and du = 2x dx. The integral becomes the integral of cos(u) du, which is sin(u) + C. Swap back to get sin(x^2) + C. The same move turns x e^(x^2) dx into (1/2) e^(x^2) + C, and u = ln x turns (ln x)/x dx into u du.

For a definite integral, carry the limits with you. Either convert them with u = g(x), so x from 0 to 1 may become u from 1 to 2, or swap back to x before you evaluate. Never evaluate x limits against a u integrand.

**Tip.** Substitution fails when the needed factor is missing. With u = x^2, du = 2x dx needs an x, so the integral of cos(x^2) dx cannot be fixed this way. Likewise e^(x^2) has no elementary antiderivative through substitution. Always check that du is really there.

**Recap.** Name the inner expression u, carry du with you, and change the limits when they exist.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A)
- [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn)

## Opens up

- [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ)
- [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf)
- [Arc Length and the Area of a Surface of Revolution](https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t)
- [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3)