Antiderivative · aiwiki.page

An antiderivative of a function $f$ is a function $F$ whose derivative equals $f$ . Also called a primitive, it reverses differentiation: given a rate of change, it reconstructs a function having that rate. Antiderivatives are central to calculus and to the evaluation of definite integrals. (math.mit.edu)

Definition and notation

For a real-valued function $f$ defined on an open interval $I$ , a differentiable function $F:I\to\mathbb{R}$ is an antiderivative of $f$ if

F'(x)=f(x)\qquad\text{for every }x\in I.

For example, $F(x)=x^3/3$ is an antiderivative of $f(x)=x^2$ , because differentiating $x^3/3$ gives $x^2$ . The process of finding an antiderivative is called antidifferentiation. (openstax.org)

The notation

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

denotes the indefinite integral, conventionally representing the family of antiderivatives on an interval. Here $C$ is an arbitrary constant, called the constant of integration. A particular antiderivative is one member of this family; an indefinite integral describes the family rather than a single numerical value. (math.mit.edu)

Antidifferentiation is not the same as finding an inverse function. An inverse function reverses the input–output action of a function, whereas an antiderivative reverses the operation of differentiation. (math.mit.edu)

Uniqueness and the constant of integration

If $F$ is an antiderivative of $f$ , then $F+C$ is also an antiderivative, because a constant has derivative zero. Conversely, if $F$ and $G$ are antiderivatives of the same function on an interval, then

(F-G)'=0.

The mean value theorem implies that $F-G$ is constant. Thus every antiderivative on that interval has the form $F+C$ . Geometrically, their graphs differ only by vertical translations. (math.mit.edu)

The interval condition matters. Applying this uniqueness result separately to the two intervals in the domain $\mathbb{R}\setminus\{0\}$ , all antiderivatives of $1/x$ have the form

F(x)= \begin{cases} \ln|x|+C_-,&x<0,\\ \ln|x|+C_+,&x>0, \end{cases}

where $C_-$ and $C_+$ need not be equal. Differentiation imposes no relation between constants on disconnected pieces of the domain. (math.mit.edu)

A prescribed value determines the constant uniquely on an interval. For instance,

F'(x)=2x,\qquad F(1)=5

gives $F(x)=x^2+4$ . This is a simple initial-value problem for a differential equation. (openstax.org)

Connection with definite integration

The fundamental theorem of calculus connects antiderivatives with accumulated quantities. If $f$ is continuous on $[a,b]$ , then

A(x)=\int_a^x f(t)\,dt

satisfies $A'(x)=f(x)$ for $a<x<b$ . Consequently, every continuous function on an interval has an antiderivative. (openstax.org)

The theorem also states that, for continuous $f$ and any antiderivative $F$ ,

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

For example,

\int_0^2 x^2\,dx =\left[\frac{x^3}{3}\right]_0^2 =\frac83.

An arbitrary constant cancels in the endpoint difference. (openstax.org)

The distinction remains important: an antiderivative is defined by a derivative equation, whereas a definite integral is defined through accumulation, such as a limit of Riemann sums. The theorem links these independently defined concepts under suitable hypotheses. Their relationship was developed by Isaac Newton, Gottfried Wilhelm Leibniz, and others during the late seventeenth and early eighteenth centuries. (openstax.org)

Existence and discontinuity

Continuity is sufficient for the existence of an antiderivative, but it is not necessary. Derivatives themselves may be discontinuous. Nevertheless, Darboux’s theorem states that every derivative has the intermediate-value property: between any two derivative values, it assumes every intermediate value. A function failing this property cannot have an antiderivative on the interval concerned. (mathweb.ucsd.edu)

For example, consider

f(x)= \begin{cases} 0,&x<0,\\ 1,&x\geq0. \end{cases}

On an interval containing zero in its interior, this function takes values $0$ and $1$ , but never $1/2$ . Darboux’s theorem therefore excludes an antiderivative there. Yet the function is Riemann integrable on every bounded interval. Its accumulated integral has a corner at zero rather than a derivative there. This illustrates why integrability alone does not guarantee an everywhere-defined classical antiderivative. (mathweb.ucsd.edu)

Basic formulas and methods

Elementary antiderivatives follow by reversing familiar differentiation rules:

\begin{aligned} \int x^p\,dx&=\frac{x^{p+1}}{p+1}+C && (p\ne-1),\\ \int \frac1x\,dx&=\ln|x|+C && (x\ne0),\\ \int e^x\,dx&=e^x+C,\\ \int \cos x\,dx&=\sin x+C,\\ \int \sin x\,dx&=-\cos x+C. \end{aligned}

For arbitrary real $p$ , the power formula is valid on $x>0$ ; other domains require $x^p$ to be appropriately defined and differentiable. Antidifferentiation also respects sums and constant multiples, allowing a polynomial to be integrated term by term. (math.mit.edu)

Substitution reverses the chain rule. If $F'=f$ , then

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

For example, setting $u=x^2$ gives

\int 2x\cos(x^2)\,dx=\sin(x^2)+C.

The factor $2x$ , the derivative of the inner function, is essential. (openstax.org)

Integration by parts reverses the product rule:

\int u(x)v'(x)\,dx =u(x)v(x)-\int u'(x)v(x)\,dx.

It can replace an integral with a simpler one. Taking $u=x$ and $v'=e^x$ , for instance, yields

\int xe^x\,dx=(x-1)e^x+C.

Any proposed antiderivative can be checked directly by differentiation. (openstax.org)

Non-elementary antiderivatives

Having an antiderivative does not imply having one expressible through elementary functions—finite expressions built from algebraic functions, exponentials, logarithms, trigonometric functions, and their inverses. Some simple integrands have no elementary antiderivative. This is a mathematical restriction, not merely a failure to find an effective substitution. (math.mit.edu)

A standard example is $e^{-x^2}$ . Its antiderivatives exist because it is continuous, but they are non-elementary. The error function is defined by

\operatorname{erf}(x) =\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt,

\int e^{-x^2}\,dx =\frac{\sqrt{\pi}}2\operatorname{erf}(x)+C.

An integral representation therefore specifies an antiderivative exactly even when elementary notation cannot express it. (math.mit.edu)

Applications

Antiderivatives recover quantities from their rates of change. In one-dimensional motion, integrating velocity produces position, while integrating acceleration produces velocity. Initial position and velocity supply the constants that differentiation cannot determine. For constant acceleration $a$ ,

v(t)=v_0+at,\qquad s(t)=s_0+v_0t+\frac12at^2.

These equations illustrate both reconstruction from a derivative and the need for initial data. (openstax.org)

More generally, an antiderivative of a continuous rate $r(t)$ determines net change:

Q(t_2)-Q(t_1) =\int_{t_1}^{t_2}r(t)\,dt.

The accumulated change is fixed by the rate, but the absolute value of $Q$ requires an additional reference value. (openstax.org)

References

1 The Anti-derivativemath.mit.edu
10 Antiderivatives - Calculus Volume 1 | OpenStaxopenstax.org
Chapter 11: The Antiderivative or Indefinite Integralmath.mit.edu
3 Uniqueness of Antiderivativesmath.mit.edu
3 The Fundamental Theorem of Calculus - Calculus Volume 1 | OpenStaxopenstax.org
Intermediate-value theorem for derivatives (Darboux’s Thm)mathweb.ucsd.edu
5 Substitution - Calculus Volume 2 | OpenStaxopenstax.org
1 Integration by Parts - Calculus Volume 2 | OpenStaxopenstax.org
5 Unintegrable Functionsmath.mit.edu
DLMF: §7.2 Definitionsdlmf.nist.gov