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===Applications===
===Applications===
The first part of the [[Fundamental Theorem of Calculus]] states that integration is the reverse function of [[derivative|differentiation]]. Thus, if <math>\int f(x) dx = g(x)</math>, then <math>\frac{d}{dx}g(x) = f(x)</math>.
The first part of the [[Fundamental Theorem of Calculus]] states that integration is the reverse function of [[Derivative (calculus)|differentiation]]. 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 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.
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.