Indefinite Integral¶

Definition¶

The set of all antiderivatives of a function $f$ is the indefinite integral of $f$ with respect to $x$ , denoted by $\displaystyle\int f(x)\,\mathrm{d}x$ . On an interval, if $F$ is an antiderivative of $f$ , then $\displaystyle\int f(x)\,\mathrm{d}x=F(x)+C$ $C$ , where $C$ is an arbitrary constant. On a union of disjoint open intervals, the constants may be chosen independently on each interval.

Interpretation¶

The indefinite integral collects all antiderivatives of $f$ in one expression. It is not a number but a family of functions, and on an interval the family is $F+C$ $C$ : one antiderivative shifted by every constant $C$ .

The notation deliberately resembles the definite integral. By the fundamental theorem of calculus, any member of the family evaluates definite integrals, $\int_a^b f(x)\,\mathrm{d}x=F(b)-F(a)$ $F (a)$ , and the constant $C$ cancels in the difference. An equation such as $\int 2x\,\mathrm{d}x=x^2+C$ is an equality of families, and it is checked by differentiating the right-hand side.

Because the answer is a family, correct answers can look different. Both $\sin^2x+C$ and $-\cos^2x+C$ are correct for $\int 2\sin x\cos x\,\mathrm{d}x$ : differentiating either gives $2\sin x\cos x$ , and $\sin^2x-(-\cos^2x)=1$ is a constant.

The single constant $C$ relies on the domain being an interval. For $\int\frac{\mathrm{d}x}{x}=\ln|x|+C$ the formula holds on $(-\infty,0)$ and on $(0,\infty)$ separately: the function equal to $\ln|x|$ for $x<0$ and to $\ln|x|+1$ for $x>0$ also has derivative $1/x$ at every $x\ne0$ $0$ .

Later, the rules of substitution and integration by parts help to find antiderivatives that cannot be read off directly.

Key Results¶

On an interval, antiderivatives differ by an additive constant.
Differentiation recovers the original integrand from any antiderivative.

Examples¶

$\displaystyle\int 2x\,\mathrm{d}x = x^2 + C\text{.}$
$\displaystyle \int \cos x\,\mathrm{d}x = \sin x + C\text{.}$

Prerequisites¶

Antiderivative

Sources¶

Math I: Calculus · Lecture 11

Aliases: antiderivative, anti-derivative, primitive function Symbols: infinity, integral

Last updated: 27 October 2026 11:52