--- title: "Integration by Substitution" description: "Use a change of variables to reverse a chain-rule pattern when the differential matches. For definite integrals, either convert the bounds or return to the original variable before evaluating. Disting" 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: 2026-11-12 --- > **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 Use a change of variables to reverse a chain-rule pattern when the differential matches. For definite integrals, either convert the bounds or return to the original variable before evaluating. Distinguish a valid substitution from a shortcut that does not simplify the integral. 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 and verify a substitution and its differential, including the intended transformed integrand - Evaluate definite substituted integrals with matching variables and bounds - Explain the limitations of a direct reverse-chain-rule shortcut without declaring every change of variables invalid ## Lesson: Rename the inside to undo the chain rule For a direct reverse-chain-rule substitution, choose an inner expression g(x) whose derivative occurs beside it, perhaps multiplied by a constant. Set u = g(x) and du = g'(x) dx. To find that derivative, use the plain derivatives: the derivative of x to a power n is n times x to the power n minus 1, the derivative of ln x is 1 over x, and the derivative of sin x is cos x. For 3x²sin(x³) dx, u = x³ changes the integrand to sin(u) du. State the transformed expression you want to obtain. More than one substitution may work. Compute the differential by differentiating the full expression for u: du = u'(x) dx. For u = 5x²+1, du = 10x dx. A constant added inside u contributes zero to its derivative. For a linear substitution u = ax+b with a nonzero, dx = du/a. **Example.** For x e^(4x²) dx, set u = 4x², so du = 8x dx and x dx = du/8. The integral becomes (1/8) integral e^u du, giving e^(4x²)/8+C. The elementary antiderivatives used here are integral e^u du = e^u+C and integral cos u du = sin u+C, with trigonometric arguments in radians. **Example.** For continuous f on [a,b], use an antiderivative F to evaluate its definite integral as F(b)-F(a). Evaluate the integral from 0 to 2 of 2x(2+x²) dx. With u = 2+x² and du = 2x dx, the bounds become 2 and 6. Integrate u to obtain u²/2, then evaluate (36-4)/2 = 16. The power rule gives integral u^n du = u^(n+1)/(n+1)+C for n ≠ -1 where the expressions are defined. Convert each bound with the actual substitution. For u = x²+5, x from 2 to 3 gives u from 9 to 14. A decreasing substitution can reverse the bound order; keep that order with its differential sign. Alternatively, back-substitute the antiderivative and use the original x bounds. Either route is valid; do not mix variables. Without a matching derivative factor, the direct shortcut may leave an extra factor. On a branch where x > 0, u = x² gives x = sqrt(u) and dx = du/(2sqrt(u)). Thus changing variables is still possible, but it need not produce a basic integral. Missing a factor alone does not prove that no elementary antiderivative exists. Check the entire transformed differential, not just the inner expression. On x > 0, u = x⁴ in e^(x⁴) dx leaves e^u du/(4u^(3/4)). The extra factor prevents reduction to integral e^u du by this direct shortcut. To check a proposed antiderivative, differentiate it and compare the full result with the integrand. **Example.** For x > 0, u = ln x has du = dx/x. The integral from e to e² of (ln x)/x dx becomes integral from 1 to 2 of u du, with value (4-1)/2 = 1.5. Powers of ln x similarly become powers of u. The reciprocal integral is integral du/u = ln|u|+C on intervals excluding zero. Using radians, u = sin x gives du = cos x dx. Thus a product of a power of sin x and cos x dx becomes the corresponding power of u. For a definite integral, evaluate sin at each endpoint before applying the u antiderivative. For an indefinite integral, return to x and include C. *(drawing: A valid change of variables may still leave an integral that the direct shortcut does not resolve.)* **Recap.** For indefinite integrals, return to x and include C. For definite integrals, convert bounds or back-substitute before evaluation. *Original integration-by-substitution lesson, questions and flashcards written for this learning dataset.* ## 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) - [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3) - [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg)