Changes

There are important differences 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,

There are important differences 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,

:<math>\frac{d}{dx}g(x)=f(x)</math>

:<math>\frac{d}{dx}F(x)=f(x)</math>

The integral of a function can be evaluated using its antiderivative,

The integral of a function can be evaluated using its antiderivative,

:<math>\int_a^b f(x)dx=g(b)-g(a)</math>

:<math>\int_a^b f(x)dx=F(b)-F(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 function]]s, [[exponential]]s, and [[logarithm]]s) and function operations.

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 function]]s, [[exponential]]s, and [[logarithm]]s) and function operations.