int.tex

\section{Integral Calculus}
\subsection{Fundamental Theorem of Calculus}
\begin{thrm}[First Fundamental Theorem of Calculus]
	\label{ftc}
	If $f$ is continuous on $[a,b]$, then the function defined by
	$$S(x)=\int _{ a }^{ x }{ f(t)\, dt }$$
	is continuous on $[a,b]$ and differentiable on $(a,b)$, and $S'(x)=f(x)$.
\end{thrm}
Written in Leibniz notation,
$$\diff{x}\int_{a}^{x} f(t)\, dt = f(x)$$

\begin{thrm}[Second Fundamental Theorem of Calculus]
	If $f$ is a continuous function on $[a,b]$, then
	$$\int _{ a }^{ b }{ f(x)\, dx=F(b)-F(a)}$$
	where $F$ is the anti-derivative of $f$, i.e. $F'=f$.
\end{thrm}
\subsection{Common Antiderivatives and Integrals}
\paragraph{Antiderivatives}
\begin{align*}
&\int\frac{1}{x}\der{x} = \ln|x| + c & &\int{\frac{1}{ax + b}\der{x}} = \frac{1}{a}\ln|ax+b| + c \\
&\int \cos a \der{x}= \frac1a \sin ax + c & &\int \sin ax\der{x}= - \frac1a \cos ax + c  &  &\int \sec^2x \der{x} =
\end{align*}