1. The problem statement, all variables and given/known data I try to solve this integral with with parameter x as a member of this scale:(-∞ , +∞) I=∫∏dx exp(-0.5XAX + XB)=∫∏dx exp( Ʃ-0.5xa[j]x[j] +Ʃ xb ) In which a[j] and b are components of telated matrix and vector and the first sum is on i and j ranges from 1 to N .Also X and B are two vector with N component and A is a N*N matrix,so the integral is over all x (which denote components of X). 2. Relevant equations Gaussian integrals in the same scale obey this equation: ∫dx exp( -ax^2 ) =√(π/a) 3. The attempt at a solution Using a change in variables like X=Y + CB with CA=Ac=1 should be appropriate: -0.5(XAX) + XB=(-0.5)(Y+CB)A(Y+CB)+(Y+CB)B=(-0.5)(YAY+YACB+CBAY+(CB)^2)+YB+CB^2 = (-0.5)(YAY+YACB+CBAY+(CB)^2)+YACB+CB^2 =(-0.5)(YAY+CBAY+(CB)^2-YACB + 2CBB) The problem is there is no YY=Ʃyy to help me change this problem to a Gaussian problem.Another problem is with YACB and CBAY that arenot compatible with this equation: (A+B)^2=A^2+B^2+AB+BA . Thank you for noticing.