1) your integral is not really a line-integral.
2) your ArcTan seems to be wrong:
define, as you do
xx[r_] := r Cos[w]; yy[r_] := r Sin[w];
then with
In[2]:= w = ArcTan[2/3];
{xx[Sqrt[2^2 + 3^2]], yy[Sqrt[13]]}
Out[3]= {3, 2}
and that is not your B.
Then if you change your parameter from t to r your Integral should read (rule of Substitution)
Integrate[f[x[t], y[t]], t] == Integrate[f[x[t[r]], y[t[r]]] D[t, r], r]
Taking into account that your t-Domain ( 0, 1 ) is mapped unto the r -Domain ( 0, Sqrt[ 13 ] ) one can say
t = r / Sqrt[ 13 ]
Hence
D[ t, r ] = 1 / Sqrt[ 13 ]
With this and the correct ArcTan you get
In[13]:= varianza = 2; w = ArcTan[3/2];
Integrate[(Sqrt[2^2 + 3^2] Exp[(-1.) ((xx[r])^2 + (yy[r])^2)/(2. varianza)]/(2 Pi*varianza)) 1/Sqrt[13],
{r, 0, Sqrt[2^2 + 3^2]}]
Out[13]= 0.139526