antiderivative

(noun)

an indefinite integral

Related Terms

Examples of antiderivative in the following topics:

  • Antiderivatives

    • An antiderivative is a differentiable function $F$ whose derivative is equal to $f$ (i.e., $F' = f$).
    • An antiderivative is a differentiable function F whose derivative is equal to $f$ (i.e., $F'=f$).
    • If $F$ is an antiderivative of $f$, and the function $f$ is defined on some interval, then every other antiderivative $G$ of $f$ differs from $F$ by a constant: there exists a number $C$ such that $G(x) = F(x) + C$ for all $x$.
    • is the most general antiderivative of $f(x) = \frac{1}{x^2}$ on its natural domain of:
    • Calculate the antiderivative (aka the indefinite integral) for a given function

  • Indefinite Integrals and the Net Change Theorem

    • As you remember from the atoms on antiderivatives, $F$ is said to be an antiderivative of $f$ if $F'(x) = f(x)$.
    • However, $F$ is not the only antiderivative.
    • For example, the function $F(x) = \frac{x^3}{3}$ is an antiderivative of $f(x) = x^2$.
    • Therefore, all the antiderivatives of $x^2$ can be obtained by changing the value of $C$ in $F(x) = \left ( \frac{x^3}{3} \right ) + C$, where $C$ is an arbitrary constant known as the constant of integration.
    • Essentially, the graphs of antiderivatives of a given function are vertical translations of each other, with each graph's location depending upon the value of $C$.

  • Inverse Trigonometric Functions: Differentiation and Integration

    • It is useful to know the derivatives and antiderivatives of the inverse trigonometric functions.
    • The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions.
    • Thus each function has an infinite number of antiderivatives.

  • The Substitution Rule

    • Integration by substitution is an important tool for mathematicians used to find integrals and antiderivatives.
    • Using the fundamental theorem of calculus often requires finding an antiderivative.
    • An antiderivative for the substituted function can hopefully be determined; the original substitution between $u$ and $x$ is then undone.
    • Similar to our first example above, we can determine the following antiderivative with this method:
    • Use $u$-substitution (the substitution rule) to find the antiderivative of more complex functions

  • Numerical Integration

    • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
    • An example of such an integrand $f(x)=\exp(-x^2)$, the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
    • It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative.
    • That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a special function which is not available.

  • The Fundamental Theorem of Calculus

    • Loosely put, the first part deals with the derivative of an antiderivative, while the second part deals with the relationship between antiderivatives and definite integrals.
    • This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.
    • The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives.

  • Numerical Integration

    • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
    • An example of such an integrand is $f(x) = \exp(x^2)$, the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
    • It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative.
    • That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a special function which is not available.

  • Iterated Integrals

    • While the antiderivatives of single variable functions differ at most by a constant, the antiderivatives of multivariable functions differ by unknown single-variable terms, which could have a drastic effect on the behavior of the function.

  • The Natural Logarithmic Function: Differentiation and Integration

    • The natural logarithm allows simple integration of functions of the form $g(x) = \frac{f '(x)}{f(x)}$: an antiderivative of $g(x)$ is given by $\ln\left(\left|f(x)\right|\right)$.

  • Area and Distances

    • Integration is connected with differentiation through the fundamental theorem of calculus: if $f$ is a continuous real-valued function defined on a closed interval $[a,b]$, then, once an antiderivative F of f is known, the definite integral of $f$ over that interval is given by$\int_{a}^{b}f(x)dx = F(b) - F(a)$.
    • Applying the fundamental theorem of calculus to the square root curve, $f(x) = x^{1/2}$, we look at the antiderivative, $F(x) = \frac{2}{3} \cdot x^\frac{3}{2}$, and simply take $F(1) − F(0)$, where $0$ and $1$ are the boundaries of the interval $[0,1]$.