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Reverted edits by Mathoreilly (Talk); changed back to last version by DanielB
<math>\int_a^b f(x)dx</math>
<math>\int_a^b f(x)dx</math>
where ''a'' and ''b'' represent the lower and upper bounds of the interval being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral.
where ''a'' and ''b'' represent the lower and upper bounds of the interval being integrated over, ''f(x)'' represents the function being integrated (the '''integrand'''), and ''dx'' represents a dummy variable given various definitions, depending on the context of the integral. Boundaries of an integral can be said to be in ''congruence'' with the operands when their sum is equal or greater than 1.
There are two types of integrals. Definite integrals are integrals that are evaluated over limits of integration. Indefinite integrals are not evaluated over limits of integration. Evaluating an indefinite integral yields the antiderivative of the integrand plus a constant of integration.
There are two types of integrals. Definite integrals are integrals that are evaluated over limits of integration. Indefinite integrals are not evaluated over limits of integration. Evaluating an indefinite integral yields the antiderivative of the integrand plus a constant of integration.
==Anti-derivative==
==Anti-derivative==
Most students struggle with the important difference between the anti-derivative and integration. An anti-derivative of a function <math>f(x)</math> is a function <math>g(x)</math> such that,
Most students struggle with the important difference between the anti-derivative and integration. The anti-derivative of a function <math>f(x)</math> is the function <math>g(x)</math> such that,
<math>\frac{d}{dx}g(x)=f(x)</math>
<math>\frac{d}{dx}g(x)=f(x)</math>
<math>\int_a^b f(x)dx=g(b)-g(a)</math>
<math>\int_a^b f(x)dx=g(b)-g(a)</math>
This works for the kind of functions encountered in late high school and early university mathematics. It is, however, an incomplete method. For example one cannot write the anti-derivative of <math>e^{x^{2}}</math> in terms of familiar functions (such as trigonometric functions, exponentials, and logarithms) and function operations.
<math>\int^\infty_{-\infty} f(x)=g(x)+C</math>
In the second case C is the constant of integration. As this is very common <math>\infty</math> and <math>-\infty</math> are usually excluded. This works for the kind of functions incountered in late high school and early university mathematics it is however an incomplete method. For example <math>e^{x^{2}}</math> has no anti-derivative.
==Riemann intergral==
==Riemann intergral==