I think this is more a mathematical question, but anyway...
In this case one is lucky, because the inner integral can be done analytically!
What remains is then just a single integral which can be calculated numerically at least. Here is my result:
In[1]:= ClearAll["Global`*"];
In[2]:= fInt[x_] = Integrate[(1 - Exp[-5.5/Cos[x]]) Sin[x], x]
Out[2]= -1. Cos[x] + E^(-5.5 Sec[x]) Cos[x] +
5.5 ExpIntegralEi[-5.5 Sec[x]]
In[3]:= NIntegrate[
fInt[ArcTan[300 Cos[y] + Sqrt[12.25 - 90000 Sin[y]^2]]] -
fInt[0], {y, 0, Pi}]
Out[3]= 3.13316 + 0. I
In[4]:= Chop[%]
Out[4]= 3.13316
Cheers Henrik