Differentiation takes a function and produces its rate of change. The indefinite integral reverses this process: given a rate of change, find the original function.
If F′(x)=f(x), then F is an antiderivative of f. The indefinite integral collects all such antiderivatives:
∫f(x)dx=F(x)+C
The constant C reflects a fundamental ambiguity. Since the derivative of any constant is zero, functions differing by a constant share the same derivative. The family F(x)+C captures every function whose derivative equals f(x).
Finding antiderivatives is the central challenge. Unlike differentiation, which follows mechanical rules, integration demands pattern recognition, technique, and sometimes ingenuity.
Antiderivatives
A function F is an antiderivative of f if:
F′(x)=f(x)
The antiderivative reverses differentiation. Given f(x)=2x, the function F(x)=x2 is an antiderivative because (x2)′=2x.
But F(x)=x2+5 is also an antiderivative—its derivative is likewise 2x. So is x2−17, or x2+π. Any function of the form x2+C differentiates to 2x.
Antiderivatives are not unique. They form a family of functions, all differing by constants.
The Constant of Integration
If F(x) is one antiderivative of f(x), then every antiderivative has the form:
F(x)+C
where C is an arbitrary constant. This follows from a basic fact: if two functions have the same derivative on an interval, they differ by a constant.
The "+C" in indefinite integrals is not optional notation—it represents the complete answer. Omitting it gives only one member of the family when infinitely many exist.
Initial conditions pin down C. If you know that F(0)=3, for instance, you can solve for the specific constant that satisfies this requirement.
Notation
The indefinite integral is written:
∫f(x)dx=F(x)+C
The integral sign ∫ without limits indicates an indefinite integral. The integrand f(x) is the function being integrated. The differential dx specifies the variable.
The result is a function (or family of functions), not a number. This contrasts with the definite integral, which produces a numerical value.
The notation mirrors the definite integral deliberately. The Fundamental Theorem of Calculus connects them: indefinite integrals provide the antiderivatives that definite integrals evaluate.
Basic Antiderivative Formulas
Several antiderivatives appear constantly and should be memorized.
Indefinite integrals obey the same linearity rules as definite integrals.
Sum rule:
∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx
Constant multiple rule:
∫c⋅f(x)dx=c∫f(x)dx
These rules reduce complex integrands to combinations of simpler ones. For example:
∫(3x2+5x−2)dx=3⋅3x3+5⋅2x2−2x+C=x3+25x2−2x+C
Verifying Antiderivatives
Integration has a built-in check: differentiate your answer.
If ∫f(x)dx=F(x)+C, then F′(x) must equal f(x). If it doesn't, an error occurred.
For example, suppose you compute:
∫sec2xdx=tanx+C
Verify: (tanx)′=sec2x. Correct.
This check catches sign errors, missing constants, and algebraic mistakes. It works because differentiation is mechanical—once you have a candidate antiderivative, verification is straightforward.
Connection to Definite Integrals
Indefinite and definite integrals serve different purposes but are linked by the Fundamental Theorem of Calculus.
The indefinite integral finds antiderivatives:
∫f(x)dx=F(x)+C
The definite integral uses an antiderivative to compute accumulated quantity:
∫abf(x)dx=F(b)−F(a)
The constant C cancels when evaluating F(b)−F(a), so any antiderivative works. This connection, detailed on the rules page, is why mastering indefinite integration enables computation of definite integrals.