how would i type $\displaystyle \int$ and include the "from x=1 to x=3" in a definite integral problem
I know you have your answer, but as I like proper spacing and you are asking about the code for integrals, I feel like noting:
In between the integrand and the $\displaystyle dx,$ you should add a \, to get the appropriate spacing.
givesCode:\int_a^bf(x)\,dx
$\displaystyle \int_a^bf(x)\,dx$
which looks nicer than
$\displaystyle \int_a^bf(x)dx$
(especially for iterated integrals).
TeX provides four spacing commands for use in math mode. they are \, (thin space), \: (medium space), \; (thick space) and \! (negative thin space). On the whole, TeX and LaTeX are pretty good at getting the spacing correct, but there are some occasions where you need to use the spacing commands to improve the appearance of a formula. One of these is the thin space before dx in an integral. Another one that I often use is a negative space (or two) after the integral sign. I think that $\displaystyle \int\!\!f(x)\,dx$ looks better than $\displaystyle \int f(x)\,dx$.