Mathematica functions start with capital letter, chnage exp ->Exp Try

Integrate[
 0.5*Exp[-90*(x - 0.5)^2 - 45*(y + 0.1)^4] + 
  Exp[-45*(x + 0.4)^2 - 60*(y - 0.5)^2], {x, -1, 1}, {y, -1, 1}]
(*0.06046*)

Mathematica functions start with capital letter, chnage exp ->Exp

Rationalize the integrand before using symbolic Integrate

Integrate[Rationalize[0.5*Exp[-90*(x - 0.5)^2 - 45*(y + 0.1)^4] + 
Exp[-45*(x + 0.4)^2 - 60*(y - 0.5)^2]], {x, -1, 1}, {y, -1, 1}] 
%//N (*0.125844*) 

NIntegrate[ 0.5*Exp[-90*(x - 0.5)^2 - 45*(y + 0.1)^4] + 
Exp[-45*(x + 0.4)^2 - 60*(y - 0.5)^2] , {x, -1, 1}, {y, -1, 1}] 
(*0.125844*)