u-substitution as the reverse Chain Rule
Replace a repeated inner expression and its derivative with one variable so the antiderivative’s structure becomes visible.
LaTeX article Updated July 11, 2026
What the substitution accomplishes
The method is not a letter swap. It packages an inside function and its differential so the integral becomes a familiar basic form.
A successful substitution removes every x. If x remains, either rewrite it using u or choose a different substitution.
Constant multiples are harmless
The derivative does not need to appear exactly. If it differs by a nonzero constant, factor that constant in or out. What matters is structural proportionality.
Do not invent a missing variable factor; constants can be adjusted, variable expressions cannot.
Definite integrals have two clean options
Either find an antiderivative in u and return to x before using the original bounds, or convert the bounds to u and stay in u. Mixing x-bounds with a u-integrand is inconsistent.
Worked example
Common mistakes
- Changing part of the integrand to u while leaving unexplained x-terms.
- Forgetting to substitute back in an indefinite integral.
- Using original x-bounds after converting the integrand to u.
Keep these ideas
- Find an inside function and its derivative.
- Adjust constants, not variable factors.
- A complete substitution uses one variable at a time.