Should you want to proceed from {0,0} to {2,3} in a nonlinear way (nonlinear means, the straight line remains straight, but is divided in non-uniform intervals), e.g.
t = r^n/Sqrt[13];
then
xx[r_] := r^n Cos[w]; yy[r_] := r^n Sin[w];
Indeed
In[3]:= w = ArcTan[3/2];
{xx[13^(1/(2 n))], yy[13^(1/(2 n))]} // PowerExpand
Out[4]= {2, 3}
and
In[5]:= varianza = 2;
integral =
Integrate[(Sqrt[2^2 + 3^2] Exp[(-1.) ((xx[r])^2 + (yy[r])^2)/(2. varianza)]/(2 Pi*varianza)) D[t, r],
{r, 0,13^(1/(2 n))}]
Out[6]= (1/(4 \[Pi]))n If[n > 0, 1.75333/n,
Integrate[E^(-0.25 r^(2 n)) r^(-1 + n), {r, 0, 13^(1/(2 n))},
Assumptions -> n <= 0]]
Looks complicated, but
In[7]:= Simplify[integral, n > 0]
Out[7]= 0.139526