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 \(b\).