Substitution rule: what it means and how to use it
Reverse the chain rule by replacing an inner expression.
Substitution rule is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why substitution rule works
The chain rule turns a composition into an outer derivative times an inner derivative. Substitution reads that sentence from right to left: if the inner derivative is already sitting in the integrand as a factor, the whole expression is something differentiated, and renaming the inside as u makes that obvious.
What each symbol means
$u=g(x)$ and $du=g'(x)dx$.
Substitution rule: when it holds
The differential factor must be present up to a constant; transform bounds in a definite integral.
When it stops applying
It needs the inner derivative to be present already, up to a constant multiple. The integral of cos of x squared has no x factor to supply the missing 2x, and no clever choice fixes that, because the integral has no elementary answer at all.
Substitution rule: a worked example
$\int2x\cos(x^2)dx=\sin(x^2)+C$ using $u=x^2$.
The mistake to avoid
What people do: Choosing a substitution and then leaving a stray x behind in the integral.
Why it goes wrong: The rewritten integral has to contain only the new variable and its differential, so a surviving x means the substitution is unfinished or the wrong piece was chosen.
Do this instead: Either solve the substitution equation for x and replace it, or pick a different inside expression so that every x disappears.
Substitution rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $u=g(x)$ and $du=g'(x)dx$.
- Check the conditions before substituting. The differential factor must be present up to a constant; transform bounds in a definite integral.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Integration
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where substitution rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Integration Techniques lesson — the lesson behind this formula: choose substitution, parts, or simplification from structure.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about substitution rule
Do I have to change the limits on a definite integral?
You either convert both limits into the new variable, or you turn the answer back into x before substituting. Mixing new limits with an old-variable expression is a classic wrong answer.
Where does the dx go?
It gets absorbed. The substitution equation gives a differential relation, and the inner-derivative factor together with dx becomes the new differential exactly.
Can I supply a missing constant myself?
Yes for numbers only. If you need 2x and only have x, put one half in front of the integral. Never try this trick with a factor that contains the variable.
How do I choose what to call u?
Look for an inside expression whose derivative is also lurking in the integrand, usually the thing under a root, up in an exponent, or wrapped in parentheses.