Short answer
u-substitution reverses the chain rule. Choose u as an inner expression whose derivative appears as a factor, rewrite the integral in u, integrate, then return to x.
- Pick u so du matches a factor already present.
- For definite integrals, changing bounds in u avoids back-substitution errors.
- Track constants carefully when du is off by a factor.
Select substitution based on structure
Substitution works best when one part of the integrand is the derivative of another part. Typical patterns include powers of linear expressions, trig composites, and exponentials with linear exponents.
If du differs by a constant multiple, factor that constant outside the integral.
Execute substitution with clean bookkeeping
Write u and du explicitly, replace all x terms in the integrand, and integrate with respect to u.
For definite integrals, convert x-bounds to u-bounds at the start and finish entirely in u.
- Do not leave mixed variables after substitution.
- Check derivative of your final antiderivative.
- Convert bounds carefully when limits are numeric.
Integrate a composite polynomial
Compute integral of 2x(x^2 + 1)^5 dx.
- Let u = x^2 + 1, so du = 2x dx.
- Rewrite integral as integral of u^5 du.
- Integrate to get u^6/6, then replace u with x^2 + 1.
Result: Antiderivative is (x^2 + 1)^6/6 + C.
Common mistakes
- Choosing u with no matching derivative factor.
- Forgetting to convert dx terms into du.
- Mixing u and x in the same intermediate expression.
- Missing constant factors when du is a multiple.
Try one
For integral cos(3x) dx, what substitution helps and why?
Use u = 3x, because du = 3 dx and the integrand is a composite cosine.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.