Alignment

$$\begin{array}{{cl}\frac{\sqrt{1-{\cos }^{2}x}}{\cos x}& =& \frac{\sqrt{{\sin }^{2}x}}{\cos x}\\ & =& \frac{\sin x}{\cos x}\\ & =& \tan x\end{array}$$

Size

Integration

$${\displaystyle \int \frac{4x^3+3}{{\left(x^4+3x\right)}^7}}dx$$ $$V=\underset{\delta x\to 0}{\lim }{\displaystyle \sum _{x=a}^{x=b}\pi y^2\delta x}$$

Vectors

$$\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$$ $$\underset{˜}{v}=3\underset{˜}{i}+4\underset{˜}{j}$$

Underscript

$$\underbrace{1+2+\dots +n}$$

Set notation

$$\mathrm{Pr}\left(A|B\right)=\frac{\mathrm{Pr}\left(A\cap B\right)}{\mathrm{Pr}\left(B\right)}$$

Functions

$$f\left(x\right)=\{\begin{array}{l}-\ln \left(-x\right),\left\{x:x<-1\right\}\\ x+1,\left\{x:-1\le x\le 0\right\}\\ e^x,\left\{x:x>0\right\}\end{array}$$

Cancelling

$${}^1\frac{\bcancel{2}}{4}\times {\frac{5}{\bcancel{10}}}_5$$