User:Coldauron/Integrals Draft

Antiderivative

For some function $f(x)$, there may exist antiderivatives. A function $F(x)$ is said to be an antiderivative of of $f(x)$ if the following is true:

${\frac {\mathrm {d} }{\mathrm {d} x}}F(x)=f(x)$

Conceptually, an antiderivative is the reverse of a derivative; if the derivative of $F(x)$ is $f(x)$, then $F(x)$ is an antiderivative of $f(x)$. Antiderivatives may or may not exist for any given function. If antiderivatives do exist, they are infinitely many. This is because any constant value can be added to a function without changing its derivative. So if $F(x)$ is an antiderivative of $f(x)$, then:

${\frac {\mathrm {d} }{\mathrm {d} x}}(F(x)+c)=f(x)$

where $c$ can take on any constant value. In the common example of velocity and position functions, velocity is the derivative of position, so position is an antiderivative of velocity. The values of position could all be changed by some constant value $c$, i.e. the starting position could be shifted, and the resulting velocity function would not be affected.

Basic Integration

The concept of the antiderivative is fundamental to the concept of the integral. In its most basic form, an integral is an operation which gives the area between a function and an axis. A graphical representation of the integral will be used to illustrate this. A simple definite integral is written like this:

$\int _{x_{1}}^{x_{2}}f(x)\mathrm {d} x$

which is pronounced "The integral from $x_{1}$ to $x_{2}$ of $f(x)$ with respect to $x$". Here, $x_{1}$ and $x_{2}$ are known as the limits of integration, and the function being integrated, $f(x)$ is known as the integrand.