Antiderivative

Definition

An antiderivative of a function $f(x)$ is another function $F(x)$ such that the derivative of $F(x)$ is equal to $f(x)$. It is also known as the indefinite integral of $f(x)$.

5 Must Know Facts For Your Next Test

1. The process of finding an antiderivative is called integration.
2. Antiderivatives are determined up to an arbitrary constant, often denoted as $C$.
3. If $F(x)$ is an antderivative of $f(x)$, then every antiderivative of $f(x)$ can be written as $F(x) + C$.
4. Common functions have standard antiderivatives, such as $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number n ≠ -1.
5. To check if a function is an antiderivative, differentiate it and see if you get back the original function.

Review Questions

• What is the relationship between a function and its antiderivative?
• Explain why every antiderivative includes an arbitrary constant.
• How would you find the antiderivative of a polynomial function?

Related terms

Definite Integral: The definite integral of a function over an interval [a, b] represents the signed area under the curve from a to b.

Fundamental Theorem of Calculus: This theorem links differentiation and integration, stating that differentiation and integration are inverse processes.

Integration by Parts: A technique used to integrate products of functions based on the product rule for differentiation.