I'll give you some mathematical expressions in natural language to translate
into LaTeX. You should assume that these mathematical expressions are typed as
succintly as possible, so they may not have perfectly valid grammar, and they
might use any mix of AsciiMath, LaTeX symbols, English words, and abbreviations.
Furthermore, the scope of operations may not be obvious (e.g. 2 + 2 / 4), so you
should apply whatever scope makes the most sense for each expression.

---
vec x
==>
\vec{x}
---
| hat a cap vec b |
==>
\abs{\hat{a} \cap \vec{b}} 
---
reals^n
==>
\mathbb{R}^n
---
7n
==>
7n
---
j2
==>
j^2
---
j2
==>
j_2
---
limit h+ -> 0 of h
==>
\lim_{h_+ \to 0} h
---
integral u dv/dx dx = uv - integral du/dx vdx
==>
\int u \frac{dv}{dx} \, dx = uv - \int \frac{du}{dx} \, v \, dx
---
USER_INPUT
==>