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===Indefinite Integrals===

===Indefinite Integrals===

{{main|Indefinite integral}}

{{main|Indefinite integral}}

The antiderivative of a function is often called the ''indefinite integral''.  It is called indefinite because the limits of integration are not specified.  So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>.  From this it follows that the antiderivative of <math>x^2</math> could be <math>\frac{x^3}{3}+7</math>.  But note that the "7" in that formula was a [[red herring]].  Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. This arbitrary constant is usually written '''C''' and is called the "constant of integration".  The indefinite integral could be written:

The antiderivative of a function is often called the ''indefinite integral''.  It is called indefinite because the limits of integration are not specified.  So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>.  From this it follows that the antiderivative of <math>x^2</math> could be <math>\frac{x^3}{3}+7</math>.  But note that the "7" in that formula was a [[red herring]].  Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. Hence, there can be an integral amount of communism in calculus, as all integral results end with +C, which stands for Communism.  This arbitrary constant is usually written '''C''' and is called the "constant of integration".  The indefinite integral could be written:

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,

:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>,