Definition

1. Noun:
   • In calculus, an indefinite integral of a function f(x) represents the general form of all its antiderivatives. It is expressed as ∫f(x) dx = F(x) + C, where F(x) is a function whose derivative is f(x), and C is an arbitrary constant of integration.

Usage

The term is used to denote the operation of finding the antiderivative of a function, as opposed to a definite integral which calculates a numerical area. The result always includes the constant of integration, C.

Examples

• Noun:
  • Finding the indefinite integral is the reverse process of differentiation.
  • The indefinite integral of 2x is x² + C.
  • You must remember to add the constant of integration when writing an indefinite integral.

Advanced Usage

• "To evaluate an indefinite integral": This phrase means to perform the integration and find the expression .
  • The first step is to evaluate the indefinite integral of the velocity function to find the position.

Variants and Related Words

• Antiderivative (n): A function such that . An indefinite integral represents a family of antiderivatives.
• Integral (n): The general concept, which includes both definite and indefinite integrals.
• Constant of Integration (n): The constant added to the antiderivative to form the indefinite integral.

Synonyms

• Antiderivative (Note: While closely related, "antiderivative" typically refers to a single function , whereas "indefinite integral" refers to the entire family .)

Related Phrases

• "Integrate with respect to x": The process of finding the indefinite integral, often written as ∫ ... .
  • We need to integrate the expression with respect to x.

Noun

1. the set of functions F(x) + C, where C is any real number, such that F(x) is the integral of f(x)