Answer
$2x+y^2+yz+z+c$
Work Step by Step
$f(x,y,z)=\int_0^x f_1 (t,0,0) dt+ \int_0^y f_2 (x,t,0) dt+\int_0^z f_3 (x,y,t) dt$
This implies that
$f(x,y,z)=\int_0^x 2 dt+ \int_0^y (2t+0) dt+\int_0^z (y+1) dt$
Thus,
$f(x,y,z)=[2t]_0^x+[t^2]_0^y[y+1](t)_0^z+c=2x+y^2+yz+z+c$
