Math 2.0

definition

Indefinite integral

Definition

Let II be an open interval and let f ⁣:I→Rf \colon I \to \mathbb{R} have an antiderivative FF on II. The indefinite integral of ff is the set of all antiderivatives of ff on II. It is written ∫f(x) dx=F(x)+C,\int f(x) \, dx = F(x) + C, where CC stands for an arbitrary real constant.

Remarks

The constant CC is the constant of integration. For example, ∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C. Some texts use the name for the function x↦∫axf(t) dtx \mapsto \int_a^x f(t) \, dt, for a fixed aa.

Explanations

L1 human

The indefinite integral collects all the antiderivatives of a function. They differ only by a constant, so one antiderivative plus CC describes them all.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionDerivative, definition, no evidence levelDerivativedefinitionAntiderivative, definition, no evidence levelAntiderivativedefinitionIndefinite integral, definition, no evidence levelIndefinite integraldefinition
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Derivative definition
  4. Antiderivative definition
  5. Indefinite integral definition

Sources

  1. Web https://en.wikipedia.org/wiki/Antiderivative, Uses and properties

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