Horizontally center fractions (and expressions) in table

Somewhat late to the party, but hopefully still useful. Rather than focus one's efforts on achieving vertical centering between successive horizontal lines, I'd recommend getting rid of the horizontal (as well as vertical) lines. This approach also lets me get rid of all [12pt] spacers.

Next, those long fraction bars look very forbidding. I think your readers will appreciate you replacing the frac{<very long numerator>{<short denominator>} expressions with frac{1}{<short denominator>}[<very long numerator>] expressions.

I'd also use a longtable environment instead of a table/tabular combination, in order to allow an automatic page break if and when needed.

Finally, do get rid of the multitude of left and right sizing directives: They accomplish nothing -- except to mess with TeX's fine horizontal spacing rules.

---

documentclass[12pt]{article}
usepackage[margin=1in]{geometry} 
usepackage{mathtools, amssymb, bm, graphicx}
usepackage{array, booktabs, longtable}
setlengthLTcapwidth{textwidth}
usepackage{float,enumitem, interval}
intervalconfig{soft open fences,separator symbol=, } 

newcommandmymat{%
begin{bmatrix*}[r]
          1 &   sin(10t) & sin^2(10t) & sin^3(10t) & sin^4(10t)\
  sin(10t) & sin^2(10t) & sin^3(10t) & sin^4(10t) & sin^5(10t)\
sin^2(10t) & sin^3(10t) & sin^4(10t) & sin^5(10t) & sin^6(10t)\
sin^3(10t) & sin^4(10t) & sin^5(10t) & sin^6(10t) & sin^7(10t)\
sin^4(10t) & sin^5(10t) & sin^6(10t) & sin^7(10t) & sin^8(10t)
end{bmatrix*}}

begin{document}
title{vspace{-2cm}HW 8}
author{John Doe}
date{today}
maketitle

begin{enumerate}[left=0pt]
item Looking at $x$, we see that $T = 0.2pi$, so $x = sin(10t)$. Therefore, 
[
bm{Phi}(t) = 
begin{bmatrix} 
1 & sin(10t) & sin^2(10t) & sin^3(10t) & sin^4(10t)
end{bmatrix}^T
]
Next,
[
bm{Phi}(t)bm{Phi}^T!(t) = mymat
]

Then, the PE condition is evaluated as
[
int_{t}^{t+T}bm{Phi}(tau)bm{Phi}^T!(tau), dtau=
int_{t}^{t+T} mymat dtau
]
The following table shows the trig integrals for various powers of $n$.
begin{longtable}{@{} >{$}l<{$} >{$displaystyle}l<{$} @{}}
caption{Trig integrals for powers of $n in {0,1,dots,8} subset mathbb{Z}$.}\
toprule
n & intsin^n(10tau),dtau\ addlinespace
midrule
0 & tau\ addlinespace
1 & -frac{1}{10}cos(10tau) \ addlinespace
2 & frac{tau}{2} - frac{1}{40}sin(20tau) \ addlinespace
3 & frac{1}{10}bigl[tfrac{1}{3}cos^3(10tau) - cos(10tau)bigr] \ addlinespace
4 & frac{1}{320}bigl[120tau - 8sin(20tau) + sin(40tau)bigr] \ addlinespace
5 & frac{1}{10}bigl[-tfrac{1}{5}cos^5(10tau) + tfrac{2}{3}cos^3(10tau)- cos(10tau) bigr] \ addlinespace
6 & frac{1}{1920}bigl[600tau + 4sin^3(20tau) - 48sin(20tau) + 9sin(40tau) bigr]\ addlinespace
7 & frac{1}{10}bigl[ tfrac{1}{7}cos^7(10tau) - tfrac{3}{5}cos^5(10tau) + cos^3(10tau) - cos(10tau) bigr] \ addlinespace
8 & frac{1}{30720}bigl[8400tau + 128sin^3(20tau) - 768sin(20tau) + 168sin(40tau) + 3sin(80tau) bigr] \addlinespace 
bottomrule
end{longtable}

With $T = 0.2pi$ and applying the integral bounds, we get:
begin{longtable}{@{} >{$}l<{$} >{$displaystyle}l<{$} @{}}
toprule
n & int_{t}^{t+T} sin^n(10tau),dtau \ addlinespace
midrule
1,3,5,7 & 0 \addlinespace
0 & 0.2pi \ addlinespace
2 & 0.1pi \ addlinespace
4 & (3/40) pi \ addlinespace
6 & (1/16)pi\ addlinespace
8 & (7/128)pi \ 
bottomrule
end{longtable}
end{enumerate}
end{document}

In addition to the excellent solution provided by @muzimuzhi regarding cellspace I have also added the option of makegapedcell -- please have a look

You could add addlinespace as a third option to get extra space with the booktabs package but the downside is you would have to remove the hline which would show gaps otherwise

I have removed some packages and also the n in the first column since unicode errors were being thrown-- I am sure you will recover

MWE

documentclass{article}
usepackage{amsmath,amsfonts}
usepackage{booktabs}
usepackage[column=o]{cellspace}%
setlengthcellspacetoplimit{5pt}
setlengthcellspacebottomlimit{5pt}
usepackage{makecell}
setcellgapes{3pt}
usepackage[american,siunitx]{circuitikz}
usepackage{mathtools}
usepackage{relsize}
usepackage[margin=1in]{geometry} 
letDeclareUSUnitDeclareSIUnit
letUSSI
DeclareUSUnitmile{mi}

begin{document}
begin{table}[!htbp]
            centering
            begin{tabular}{|oc|oc|}
                hline
                & $mathlarger{intsin^nleft(10tauright)dtau}$\
                hline
                0 & $tau$\
                hline
                1 & $-dfrac{cosleft(10tauright)}{10}$\
                hline
                2 & $dfrac{tau}{2} - dfrac{sinleft(20tauright)}{40}$\
                hline
            end{tabular}
end{table}
begin{table}[!htbp]            
            centeringmakegapedcells
            begin{tabular}{|c|c|}
                hline
                & $mathlarger{intsin^nleft(10tauright)dtau}$\
                hline
                0 & $tau$\
                hline
                1 & $-dfrac{cosleft(10tauright)}{10}$\
                hline
                2 & $dfrac{tau}{2} - dfrac{sinleft(20tauright)}{40}$\
                hline
            end{tabular}
end{table}
end{document} 

A cellspace package solution, in which

• all [12pt] used in \[12pt] is deleted, since this only adds vertical space to bottom of tabular cells.
• cellspace is loaded and two length changed

documentclass{article}
usepackage{amsfonts}
usepackage{amsmath}
usepackage{float}
usepackage{relsize}

usepackage[column=O]{cellspace}
setlength{cellspacetoplimit}{5pt}
setlength{cellspacebottomlimit}{5pt}

begin{document}
begin{table}[H]
  centering
  begin{tabular}{|Oc|Oc|}
      hline
      $n$ & $mathlarger{intsin^nleft(10tauright)dtau}$\ 
      hline
      0 & $tau$\
      hline
      1 & $-dfrac{cosleft(10tauright)}{10}$\
      hline
      2 & $dfrac{tau}{2} - dfrac{sinleft(20tauright)}{40}$\
      hline
      3 & $dfrac{frac{cos^3left(10tauright)}{3} - cosleft(10tauright)}{10}$\
      hline
      4 & $dfrac{120tau - 8sinleft(20tauright) + sinleft(40tauright)}{320}$\
      hline
      5 & $dfrac{-frac{cos^5left(10tauright)}{5} + frac{2cos^3left(10tauright)}{3}- cosleft(10tauright)}{10}$\
      hline
      6 & $dfrac{600tau + 4sin^3left(20tauright) - 48sinleft(20tauright) + 9sinleft(40tauright)}{1920}$\
      hline
      7 & $dfrac{frac{cos^7left(10tauright)}{7} - frac{3cos^5left(10tauright)}{5} + cos^3left(10tauright) - cosleft(10tauright)}{10}$\
      hline
      8 & $dfrac{8400tau + 128sin^3left(20tauright) - 768sinleft(20tauright) + 168sinleft(40tauright) + 3sinleft(80tauright)}{30720}$\
      hline
  end{tabular}
  caption{Trig integrals for powers of $n in [0,8] in mathbb{Z}$.}
end{table}
end{document}