Could someone tell me how the volume-translated Peng-Robinson exactly works? How do I calculate the fugacity expression of VTPR? Do I integrate the V + c terms against dV or do i integrate it entirely with respect to d(V + c)?

According to this paper (Tsai, J-C., Chen, Y-P.: Application of a volume-translated Peng-Robinson equation of state on vapor-liquid equilibrium calculations, 1997):

https://ibb.co/fGeMhe

If by comparison the fugacity equation is the same with the original PR EOS then I assume that they integrated the volume-translated equation with respect to d(V+c)? What I did to verify is to numerically integrate the volume-translated PREOS and the result did not equal the result of the integrated equation (did it in MATLAB, as seen here)

Or am I misunderstanding something? Please help. Thank you!

https://ibb.co/k1Gv8K