Doc 4.4.1 here

Example 1

Let $f\colon[a,b]\to\R$ be Riemann integrable. Let $F\colon[a,b]\to\R$ be $F(x)=\int_{a}^{x} f(t)\,dt$. Then $F$ is continuous, and at all $x$ such that $f$ is continuous at $x$, $F$ is differentiable at $x$ with $F'(x)=f(x)$.

Example 2

$$I = \int_0^{2\pi} \sin(x)\,dx$$

Example 3

$$\begin{equation} \begin{split} a &=b+c\\ &=e+f \end{split} \end{equation}$$

Example 4

$$x = \begin{cases} a &\text{if } b \\ c &\text{if } d \end{cases}$$

Example 5

$$\overbrace{a+b+c}^{\text{up note}}$$

Example 6

$$\underbrace{a+b+c}_{\text{down note}}$$

Example 7

$$\boxed{\pi=\frac c d}$$

Example 8

$$\begin{cases} a &\text{if } b \\ c &\text{if } d \end{cases}$$

Example 9

$$\begin{CD} A @>a>> B \\ @VbVV @AAcA \\ C @= D \end{CD}$$

example 10

$$\ce{C6H5-CHO}$$

example 11

$$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}$$

example 13

$$\footnotesize footeris$$

example 14

$$\frak{CrEaTiViTaS}$$

example 15

$$\begin{gather} a=b \\ e=b+c \end{gather}$$

example 16

$$\begin{gathered} a=b \\ e=b+c \end{gathered}$$

example 17

$$\gdef\sqr#1{#1^2} \sqr{y} + \sqr{y}$$

example 18

$$\genfrac ( ] {2pt}{0}a{a+1}$$

example 19

$$\begin{matrix} a & b \\ \hdashline c & d \end{matrix}$$

example 20

$$\begin{matrix} a & b \\ \hline c & d \end{matrix}$$

example 21

$$P\impliedby Q$$

example 22

$$\begin{alignat}{2} 10&x+&3&y=2\\ 3&x+&13&y=4 \end{alignat}$$

example 23

$$\begin{gather} a=b \\ e=b+c \end{gather}$$

example 24

$$\begin{equation} \begin{split} a &=b+c\\ &=e+f \end{split} \end{equation}$$

example 25

$$\Alpha \Beta \Gamma \Delta \Epsilon \Zeta \Eta \Theta \Iota \Kappa \Lambda \Mu \Nu \Xi \Omicron \Pi \Rho \Sigma \Tau \Upsilon \Phi \Chi \Psi \Omega \varGamma \varDelta \varTheta \varLambda \varXi \varPi \varSigma \varUpsilon \varPhi \varPsi \varOmega \alpha \beta \gamma \delta \epsilon \zeta \eta \theta \iota \kappa \lambda \mu \nu \xi \omicron \pi \rho \sigma \tau \upsilon \phi \chi \psi \omega \varepsilon \varkappa \vartheta \thetasym \varpi \varrho \varsigma \varphi \digamma$$

example 26

$$\sum_{\mathclap{1\le i\le j\le n}} x_{ij}$$

example 27

$${a+1 \over b+2}+c$$

example 28

$$\overbrace{x+⋯+x}^{n\text{ times}}$$

example 29

$$\phase{-78^\circ}$$

example 30

$$\Gamma^{\phantom{i}j}_{i\phantom{j}k}$$

example 31

$$\tag{3.1c} a^2+b^2=c^2$$

example 32

$$\Set{ x | x<\frac 1 2}$$

example 33

$$a+\left(\vcenter{\hbox{$\frac{\frac a b}c$}}\right) TeX (strict) syntax$$