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→Integration: Several things have come up today that require attention, before I forget them.
Integration is the inverse function of [[differentiation]]; the two notions are related by the [[Fundamental Theorem of Calculus]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.
Integration is the inverse function of [[differentiation]]; the two notions are related by the [[Fundamental Theorem of Calculus]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.
Note that, because the derivative of any real number is 0, <math>\frac{d}{dx}x^2+2 = \frac{d}{dx}x^2+3 = \frac{d}{dx}x^2 = 2x</math>. Therefore, <math>\int 2x dx</math> does not simply equal x<sup>2</sup>, but rather x<sup>2</sup> + C for any constant real number C. The above-mentioned functions are the ''family of antiderivatives'' for 2x.
Note that, because the derivative of any constant function is 0, <math>\frac{d}{dx}x^2+2 = \frac{d}{dx}x^2+3 = \frac{d}{dx}x^2 = 2x</math>. Therefore, <math>\int 2x dx</math> does not simply equal x<sup>2</sup>, but rather x<sup>2</sup> + C for any constant real number C. The above-mentioned functions are the ''family of antiderivatives'' for 2x.
The concept of integration can be extended to functions in more than one real variable, as well as functions defined over the [[complex numbers]].
The concept of integration can be extended to functions in more than one real variable, as well as functions defined over the [[complex numbers]].