Integration and Antiderivatives

Undoing the Derivative

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

A key subtlety: antiderivatives are not unique. Since the derivative of any constant is zero, if $F$ is an antiderivative then so is $F + C$ for any constant $C$. So the indefinite integral carries a constant of integration: $$\int f(x)\, dx = F(x) + C.$$ Forgetting the $+C$ is the classic omission — the constant encodes that infinitely many functions share the same derivative, differing only by a vertical shift.

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 involved integrals use techniques like substitution (reversing the chain rule) and integration by parts (reversing the product rule), but the power rule plus the standard forms handle a great many cases.

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 integral power rule raises the exponent and divides ($\int x^n = \frac{x^{n+1}}{n+1}$) — the reverse of the derivative rule; using the derivative pattern (bringing the power down) for integration is a common slip.