Indefinite integral

Definition

An indefinite integral, also known as an antiderivative, is a function whose derivative is the given function. It represents a family of functions that differ by a constant.

5 Must Know Facts For Your Next Test

1. The general form of an indefinite integral is $\int f(x) \, dx = F(x) + C$, where $F(x)$ is the antiderivative of $f(x)$ and $C$ is the constant of integration.
2. Indefinite integrals do not have limits of integration, unlike definite integrals.
3. To find an indefinite integral, you need to determine the original function from its derivative.
4. Common techniques for finding indefinite integrals include substitution and integration by parts.
5. The power rule for integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$.

Review Questions

• What is the indefinite integral of $\cos(x)$?
• How do you apply the power rule to find the indefinite integral of $x^4$?
• Explain why constants are added to indefinite integrals.

Related terms

Definite Integral: A type of integral with upper and lower limits, representing the area under a curve between two points.

Fundamental Theorem of Calculus: Connects differentiation and integration, stating that differentiation and integration are inverse operations.

$u$-substitution: A method used to simplify integrals by substituting part of the integral with a new variable.