Antiderivatives and Indefinite Integrals

Undoing the Derivative

Integration, in one of its two guises, reverses differentiation. An antiderivative of $f$ is a function $F$ with $F'(x) = f(x)$. Where differentiation asks "what is the rate of change?", antidifferentiation asks "what function has this rate of change?" — recovering position from velocity, or a total from a rate.

A key subtlety: antiderivatives are not unique. Because the derivative of any constant is zero, if $F$ is an antiderivative then so is $F + C$ for any constant $C$. The indefinite integral therefore carries a constant of integration: $$\int f(x)\, dx = F(x) + C.$$ Omitting the $+C$ is the classic mistake — the constant encodes that infinitely many functions (a whole vertical family of parallel curves) share the same derivative.

The rules reverse the differentiation rules:

• Power rule for integration — $\int x^n\, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$): raise the exponent by one and divide by the new exponent — the opposite of the derivative power rule.
• Constant multiple and sum rules carry over: constants factor out, and the integral of a sum is the sum of the integrals.
• Reversing standard derivatives: $\int \cos x\, dx = \sin x + C$, $\int e^x\, dx = e^x + C$, and $\int \frac{1}{x}\, dx = \ln|x| + C$ (the $n = -1$ case the power rule excludes).

More elaborate integrals use substitution (reversing the chain rule) and integration by parts (reversing the product rule), covered later, but the power rule and standard forms handle a great many.

Common pitfall: forgetting the constant of integration $+C$, and misapplying the integration power rule. Every indefinite integral needs $+C$, because antiderivatives are unique only up to a constant. And the integration power rule raises the exponent and divides ($\int x^n = \frac{x^{n+1}}{n+1}$) — the reverse of the derivative rule; accidentally bringing the power down (the derivative pattern) is a common slip.