

\begin{center}
{\LARGE Antiderivative Formulas}
\end{center}

\noindent
\large
We will use the notation $\int f(x) dx$ to denote the antiderivative of $f(x)$.
In the following formulas, $f(x)$ and $g(x)$ are functions
and $c$ is a constant:

\begin{eqnarray*}
\int f + g dx & = & \int f + \int g\\
\int f - g dx & = & \int f - \int g\\
\int cf dx & = & c \int f dx\\
\int f'(g(x))g'(x) & = & f(g(x)) + C\\
\end{eqnarray*}

\noindent
Formulas for common functions:

\vspace{.5cm}
\begin{tabular}{ll}
\vspace{.1cm}
$\int x^a\ dx = \frac{x^{a+1}}{a+1} + C \mbox{\ if\ } a \neq -1$ & $\int x^{-1}\ dx = \ln x + C$\\
$\int a^x\ dx = \frac{a^x}{\ln a} + C$ & $\int \ln x\ dx = x \ln x - x + C$\\
$\int \sin x\ dx = -\cos x + C$ & $\int \cos x\ dx = \sin x + C$\\
$\int \tan x\ dx = -\ln |\cos x| + C$ & $\int \cot x\ dx = \ln |\sin x| + C$\\
$\int \sec x\ dx = \ln |\sec x + \tan x | + C$ & $\int \csc x\ dx = \ln |\csc x - \cot x| + C$\\
$\int \sec^2 x\ dx = \tan x + C$ & $\int \csc^2 x\ dx = -\cot x + C$\\
$\int \sec x \csc x\ dx = \sec x + C$ & $\int \csc x \cot x\ dx = -\csc x + C$\\
$\int \frac{1}{\sqrt{1-x^2}}\ dx = \arcsin x + C$ & $\int \frac{1}{x^2+1}\ dx = \arctan x + C$\\
$\int \frac{1}{x\sqrt{x^2-1}}\ dx = \mbox{\ arcsec x\ } + C$\\
$\int \sinh x\ dx = \cosh x + C$ & $\int \cosh x\ dx = \sinh x + C$\\
\end{tabular}

