Does an antiderivative always exist?

For any such function, an antiderivative always exists except possibly at the points of discontinuity. There is always an answer (there is always a function whose derivative is the function given to you, provided it is continuous).

Can a function not have an antiderivative?

Most functions you normally encounter are either continuous, or else continuous everywhere except at a finite collection of points. For any such function, an antiderivative always exists except possibly at the points of discontinuity.

Can different functions have the same antiderivative?

Yes,more than one function can be antiderivatives of the same function.

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

Antiderivatives are the opposite of derivatives. An antiderivative is a function that reverses what the derivative does. One function has many antiderivatives, but they all take the form of a function plus an arbitrary constant. Antiderivatives are a key part of indefinite integrals.

What functions have 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)….Exercise 6.

Function	General antiderivative	Comment
(ax+b)n	1a(n+1)(ax+b)n+1+c	for a,b,c,n any real constants with a≠0, n≠−1

For any such function, an antiderivative always exists except possibly at the points of discontinuity. For more exotic functions without these kinds of continuity properties, it is often very difficult to tell whether or not an antiderivative exists.

How do you find the general antiderivative of a function?

Using the previous example of F ( x) = x 3 and f ( x) = 3 x 2, you find that . The indefinite integral of a function is sometimes called the general antiderivative of the function as well. Example 1: Find the indefinite integral of f ( x) = cos x . Example 2: Find the general antiderivative of f ( x) = –8.

What are the antiderivatives of f x = 3×2?

For example, F ( x) = x 3, G ( x) = x 3 + 5, and H ( x) = x 3 − 2 are all antiderivatives of f ( x) = 3 x 2 because F ′ ( x) = G ′ ( x) = H ′ ( x) = f ( x) for all x in the domain of f. It is clear that these functions F, G, and H differ only by some constant value and that the derivative of that constant value is always zero.

Is there always an answer to the derivative of a function?

There is always an answer (there is always a function whose derivative is the function given to you, provided it is continuous). However, it may not be possible to express the answer in terms of familiar functions and operations. For example, the antiderivative of