Indefinite Integrals

Antiderivatives

The Constant of Integration

Indefinite Integral Notation

Basic Antiderivative Formulas

Linearity of Indefinite Integrals

Verifying Antiderivatives

Connection to Definite Integrals

Summary: The Indefinite-Integration Workflow

Indefinite Integrals FAQ

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.

Indefinite Integral Notation

Notation

Indefinite Integral Notation

The three-part symbol, the constant that is not optional, and the case convention that carries the whole subject.

dx

as a differential — differentials; the prime — the derivative function.

\int f(x)\, dx

The indefinite integral of f of x, d x

Leibniz's elongated S, for summa. No limits on the sign — indefinite: the output is a family of functions, not a number.

f(x)

is the integrand;

dx

names the variable and closes it off. The mathematical keyboard carries

\int

CasesWith limits attached it becomes the definite integral — a number, a different object, same glyph. Under substitution the

dx

is a live differential:

du = g'(x)\,dx

is real algebra, not bookkeeping.

Also written

\int dx\, f(x)

— differential first, standard in physics texts, where the variable announces itself before a long integrand.

Do not confuse

dx

as decoration. Dropping it leaves the variable unnamed —

\int 2t\,dx

and

\int 2t\,dt

are different answers.

F(x) + C

F of x, plus C

The general antiderivative. “

+C

” is part of the answer, not a garnish — The Constant of Integration above insists on it. One symbol, ranging over all reals, carries the entire family.

CasesSums take a single

C

— the pieces' constants merge on arrival. For

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

the honest version has a separate constant on each side of

0

; the single

C

is the standard abbreviation.

Also written

K

, or

c_1

, when

C

is spoken for — any letter declared arbitrary.

Do not confuseThe critical number

c

of graph analysis — one fixed, found point; this

C

is deliberately free. In definite integrals it cancels, which is why it never appears there.

F'(x) = f(x)

F prime equals f

The case convention: a capital letter names an antiderivative of its own lowercase. The letter case is the claim — writing

F

asserts

F' = f

with no further words.

Cases

G/g

H/h

work identically. The pairing carries into the Fundamental Theorem,

\int_a^b f(x)\,dx = F(b) - F(a)

— Connection to Definite Integrals below.

Also written

F = \int f

— operator shorthand, bare sign and no variables, common on blackboards.

Do not confuse“

F

is the antiderivative.”

F

is one member — the family is

F + C

, and uniqueness fails by exactly a constant.

All calculus symbols →↑ Integrals

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)

Indefinite Integrals FAQ

Why do all antiderivatives of a function differ by a constant?

Because two functions with the same derivative on an interval can only differ by a constant. If F′ = G′ then (F − G)′ = 0, and a function whose derivative is zero everywhere on an interval is constant. So once you find one antiderivative F, the whole family is F + C — nothing outside that form can have the right derivative.Read more →

Do you need a separate + C for each term?

No — one constant covers the whole answer. Integrating term by term, each piece contributes its own arbitrary constant, but a sum of arbitrary constants is just another arbitrary constant, so they merge into a single C. Writing C₁ + C₂ + C₃ is not wrong, only redundant. Collapse them and write + C once, at the end.Read more →

How do you find the value of C?

You need one known value of the antiderivative — an initial condition. Integrate to get F(x) + C, substitute the known point, and solve. If ∫2x dx = x² + C and you are told F(0) = 3, then 0² + C = 3, so C = 3 and F(x) = x² + 3. Without such a condition, C stays arbitrary.Read more →

Does the dx in an integral actually matter?

Yes — it names the variable of integration, and dropping it leaves the expression ambiguous. ∫2t dx and ∫2t dt have different answers: the first treats t as a constant and gives 2tx + C, the second integrates in t and gives t² + C. The dx is also a live differential under substitution.Read more →

Why is an antiderivative written with a capital F?

It is a case convention that carries a claim: writing F for an antiderivative of f asserts F′ = f without saying so in words. The same pairing works for G and g, or H and h. It is what makes the Fundamental Theorem readable as F(b) − F(a) — the capital signals which function is the antiderivative.Read more →