Does every elementary function have an antiderivative?

Elementary functions always have elementary derivatives. However, many elementary functions do not have an elementary antiderivative. Some of them are even quite simple looking, and many are actually useful.

What is an elementary antiderivative?

In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field …

Can a function be an antiderivative of two different functions?

Suppose A(x) and B(x) are two different antiderivatives of f(x) on some interval [a, b]. Thus any two antiderivative of the same function on any interval, can differ only by a constant. The antiderivative is therefore not unique, but is “unique up to a constant”.

Can you find the antiderivative of every function?

So, F(x) is an antiderivative of f(x). And, the theory of definite integrals guarantees that F(x) exists and is differentiable, as long as f is continuous. For any such function, an antiderivative always exists except possibly at the points of discontinuity.

Do all elementary functions have elementary derivatives?

Since every elementary function must belong to one of the categories above, it must have an elementary derivative.

Is ln an elementary function?

[edit] Logarithm The second transcendental that is considered elementary is the inverse of the exponential function, the logarithm. The logarithm is denoted ln(x).

How do you know if a function is elementary?

In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or …

How do you tell if a function has an antiderivative?

An antiderivative of a function f(x) is a function whose derivative is equal to f(x). That is, if F′(x)=f(x), then F(x) is an antiderivative of f(x).

Can you have multiple antiderivatives?

Every continuous function has an antiderivative, and in fact has infinitely many antiderivatives. Two antiderivatives for the same function f(x) differ by a constant.

How is the antiderivative of a function related to the function?

An antiderivative of a function f(x) is a function whose derivative is equal to f(x). That is, if F′(x)=f(x), then F(x) is an antiderivative of f(x). It is not one function but a family of functions, differing by constants; and so the answer must have a ‘+ constant’ term to indicate all antiderivatives.