Answer
$2$
Work Step by Step
$I=\int_{C_1}(y+z) dx+(x+z) dy+(x+y) dz+\int_{C_2}(y+z) dx+(x+z) dy+(x+y) dz$
$=\int_{0}^{1} (0+t) dt +(x+z)(0) +(t+0)dt+\int_{0}^{1} [t+(1+t)](-dt) +[(-t+1)+(1+t)] dt +[(-t+1)+t]dt$
$=\int_0^1 2t dt+\int_{0}^{1} (-2t-1) dt+2 dt+1 dt$
$=[t^2]_0^1+[-t^2+2t]_{0}^{1}$
$=(1)^2-0+1$
$=2$
