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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''.  (Indefinite because the limits a and b haven't been 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  

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:

<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:

:<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>,

The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b.  Note that the constant of integration cancels out.

The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b.  Note that the constant of integration cancels out.

===Definite Integration===

===Definite Integrals===

{{main|Definite integral}}

{{main|Definite integral}}

This kind of integral is called a ''definite integral'', written with the limits <math>[a,b]</math>:

This kind of integral is called a ''definite integral'', written with the limits <math>[a,b]</math>:

The above is a simplified "intuitive" treatment of calculus and of this theorem.  The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the [[Fundamental Theorem of Calculus]].

The above is a simplified "intuitive" treatment of calculus and of this theorem.  The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the [[Fundamental Theorem of Calculus]].

==Derivatives==

==Derivatives==