# Integral

The inverse of the derivative in the sense that the derivative of the integral of an integrable function is the original function
\begin{align*} \mathrm{if} & \; F = \int f(x) \, \mathrm{d}x\\ \mathrm{then} & \; \frac{\mathrm{d}F}{\mathrm{d}x} = f \end{align*}
Here, $$f$$ is known as the integrand.
Note that the reverse doesn't hold since the derivative of a constant is zero so adding one to $$F$$ doesn't make a difference
\begin{align*} \mathrm{if} & \; \frac{\mathrm{d}F}{\mathrm{d}x} = f\\ \mathrm{then} & \; F = \int f(x) \, \mathrm{d}x + c \end{align*}
for some value $$c$$ known as the constant of integration.

If we take the difference between the integral calculated at two points then the constant cancels out yielding a uniquely defined result written as
$\int_a^b f(x) \, \mathrm{d}x = \bigg[F(x)\bigg]_a^b = F(b) - F(a)$
The former is known as the indefinite integral and the latter as the definite integral.
The latter is also equal to the area under the curve defined by $$f$$ between the points $$a$$ and $$b$$.

### Gallimaufry

 AKCalc ECMA Endarkenment Turning Sixteen

This site requires HTML5, CSS 2.1 and JavaScript 5 and has been tested with

 Chrome 26+ Firefox 20+ Internet Explorer 9+