# 19 Integration

At one level integration may be seen as the inverse process of differentiation. We would write:

$I = \int f'(x) dx = F(x) + C$

The $+C$ is called the constant of integration. If you differentiate a constant you get zero. If you integrate an expression there may have been a constant value so we add $+C$ until we can evaluate the constant by using the boundary conditions.

## 19.1 Table of Common Standard Integrals

In the following table the left hand column contains functions of $x$ ($y=f(x)$). The right hand column contains the integrals of the functions with respect to $x$ ($\int f(x) dx=F(x)$).

$y=f(x)$    $\int f(x) = F(x)$
$x^n$$\frac{x^{n+1}}{n+1}+C x^{-1}$$ ln(x)+C$ for $x>0$
$ln(-x)+C$ for $x<0$
$cos(ax)$$sin(ax)/a+C sin(ax)$$-cos(ax)/a+C$
$tan(ax)$$ln(sec(ax))/a+C e^{ax}$$e^{ax}/a+C$