What is an antiderivative?

A function F is an antiderivative of f if F'(x) = f(x). Antiderivatives reverse differentiation. They are not unique—if F is an antiderivative of f, then so is F + C for any constant C, forming a family of functions.

What does indefinite integral notation mean?

The notation ∫f(x) dx = F(x) + C means the indefinite integral. The ∫ symbol without limits indicates an indefinite integral, f(x) is the integrand, and dx specifies the variable. The result is a function family, not a number.

What are the basic antiderivative formulas?

Key formulas include: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1; ∫1/x dx = ln|x| + C; ∫eˣ dx = eˣ + C; ∫cos x dx = sin x + C; ∫sin x dx = -cos x + C. These should be memorized.

What are the linearity rules for indefinite 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 let you break complex integrands into simpler pieces.

How are indefinite and definite integrals related?

The Fundamental Theorem of Calculus connects them. Indefinite integrals find antiderivatives: ∫f(x) dx = F(x) + C. Definite integrals evaluate: ∫ₐᵇ f(x) dx = F(b) − F(a). The constant C cancels when computing F(b) − F(a).

Indefinite Integrals

Q: Why do indefinite integrals have +C?

The constant of integration C represents the complete family of antiderivatives. Since the derivative of any constant is zero, functions differing by a constant share the same derivative. Omitting +C gives only one member when infinitely many exist.

Q: How do you verify an antiderivative?

Differentiate your answer. If ∫f(x) dx = F(x) + C, then F'(x) must equal f(x). This check catches sign errors, missing constants, and algebraic mistakes because differentiation is mechanical and straightforward.

Antiderivatives

The Constant of Integration

Notation

Basic Antiderivative Formulas

Linearity of Indefinite Integrals

Verifying Antiderivatives

Connection to Definite Integrals

Summary: The Indefinite-Integration Workflow

Reversing Differentiation

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:

\int 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.

Key Terms

Antiderivative— a function whose derivative is

f

Indefinite Integral—

\int f(x)\,dx = F(x) + C

, the full family

Integrand— the function

f(x)

being integrated

Definite Integral— connected by the Fundamental Theorem

See All Calculus Definitions →

The Constant of Integration

F(x)

is one antiderivative of

f(x)

, then every antiderivative has the form:

Antiderivative Family

\int f(x)\, dx = F(x) + C \quad \text{where } F'(x) = f(x)

Learn more about this formula: Antiderivative Family →

Browse all integral formulas →

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:

\int f(x)\, dx = F(x) + C

The integral sign

\int

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.

See All Calculus Symbols and Notations →

Basic Antiderivative Formulas

Several antiderivatives appear constantly and should be memorized.

Power rule (for

n \neq -1

\int x^n\, dx = \frac{x^{n+1}}{n+1} + C

Reciprocal:

\int \frac{1}{x}\, dx = \ln|x| + C

Exponential:

\int e^x\, dx = e^x + C

Trigonometric:

\int \cos x\, dx = \sin x + C \qquad \int \sin x\, dx = -\cos x + C

The special integrals page provides a more complete list.

Integrand f(x)	∫ f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹ / (n + 1) + C
1 / x	ln \|x\| + C
e^x	e^x + C
cos x	sin x + C
sin x	−cos x + C

Linearity of Indefinite Integrals

Indefinite integrals obey the same linearity rules as definite integrals.

Sum rule:

\int [f(x) + g(x)]\, dx = \int f(x)\, dx + \int g(x)\, dx

Constant multiple rule:

\int c \cdot f(x)\, dx = c \int f(x)\, dx

These rules reduce complex integrands to combinations of simpler ones. For example:

\int (3x^2 + 5x - 2)\, dx = 3 \cdot \frac{x^3}{3} + 5 \cdot \frac{x^2}{2} - 2x + C = x^3 + \frac{5x^2}{2} - 2x + C

Verifying Antiderivatives

Integration has a built-in check: differentiate your answer.

If

\int 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:

\int \sec^2 x\, dx = \tan x + C

Verify:

(\tan x)' = \sec^2 x

. 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:

\int f(x)\, dx = F(x) + C

The definite integral uses an antiderivative to compute accumulated quantity:

\int_a^b f(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.

Summary: The Indefinite-Integration Workflow

Every indefinite integral on this page follows the same five-step procedure, whether the integrand is a single power function or a complicated composition that requires a technique. The table below collects the workflow with a pointer to the sibling page where each step's detail lives — special integrals for the formula library, rules for linearity and the FTC, techniques for transformations. Use this as a reminder while working a problem: every step matters, and skipping the final verification is the single most common source of errors.

Step	Action	Where the detail lives
1	Recognize the form — does the integrand match a known antiderivative formula?	special integrals
2	Apply linearity — split sums and factor out constants to reduce to known pieces	integration rules
3	Compute the antiderivative F(x) — apply a formula directly, or use a technique to transform the integrand first	techniques
4	Add the constant of integration — the answer is F(x) + C, never just F(x)	obj2 above (constant of integration)
5	Verify by differentiating — F'(x) should return the original integrand f(x)	obj6 above (verifying antiderivatives)