Power antiderivative: what it means and how to use it
Reverse the power rule to find an antiderivative.
Power antiderivative is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why power antiderivative works
Differentiating x to the n+1 hands back n+1 times x to the n, so dividing by n+1 cancels that unwanted factor and returns exactly the function you started with. The constant is there because sliding a graph up or down never changes any of its slopes.
What each symbol means
$n$ is constant and $C$ represents every constant antiderivative.
Power antiderivative: when it holds
$n\ne-1$; when $n=-1$, use $\int dx/x=\ln|x|+C$.
When it stops applying
An exponent of −1 breaks it outright, because n+1 becomes 0 and the rule asks you to divide by zero. The antiderivative of 1 over x is the natural logarithm of |x| plus a constant, which is not a power at all.
Power antiderivative: a worked example
$\int x^3dx=x^4/4+C$.
The mistake to avoid
What people do: Leaving the constant of integration off an indefinite integral.
Why it goes wrong: Every function of that form plus any constant has the same derivative, so an answer without the constant names only one member of an infinite family.
Do this instead: Write the constant the moment the integral is done. In a differential equation it is exactly what the initial condition pins down.
Power antiderivative: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $n$ is constant and $C$ represents every constant antiderivative.
- Check the conditions before substituting. $n\ne-1$; when $n=-1$, use $\int dx/x=\ln|x|+C$.
- 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 power antiderivative 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 power antiderivative
Does the rule handle a square root?
Yes. Write the root as the one half power, raise it to three halves, and divide by three halves, giving two thirds of x to the three halves.
How do I integrate a plain number like 7?
Treat it as 7 times x to the zero. The exponent climbs to 1 and you divide by 1, so the antiderivative is 7x plus a constant.
Why does the exponent climb here when a derivative sends it down?
Integration runs the derivative backwards, so every step reverses: add one to the exponent rather than subtracting, and divide rather than multiply.
Can two different-looking answers both be right?
Yes, as long as they differ only by a constant. Differentiating both is the quickest way to confirm that they describe the same family.