Parameterize the first line segment [tex]C_1[/tex] by
[tex]\vec r(t)=(1,0,1)(1-t)+(2,4,1)t=(1+t,4t,1)[/tex]
and the second line segment [tex]C_2[/tex] by
[tex]\vec s(t)=(2,4,1)(1-t)+(2,6,4)t=(2,4+2t,1+3t)[/tex]
both with [tex]0\le t\le1[/tex]. Then
[tex]\displaystyle\int_{C_1}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz[/tex]
[tex]\displaystyle=\int_0^1\bigg((1+5t)\cdot1+2(1+t)\cdot4+(1-t)4t\cdot0\bigg)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^1(9+13t)\,\mathrm dt=\frac{31}2[/tex]
and
[tex]\displaystyle\int_{C_2}(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz[/tex]
[tex]\displaystyle=\int_0^1\bigg((2+(4+2t)(1+3t))\cdot0+2(2)\cdot2+2(4+2t)(1+3t)\cdot3\bigg)\,\mathrm dt[/tex]
[tex]\displaystyle=\int_0^1(32+84t+36t^2)\,\mathrm dt=86[/tex]
Then
[tex]\displaystyle\int_C(x+yz)\,\mathrm dx+2x\,\mathrm dy+xyz\,\mathrm dz=\frac{31}2+86=\boxed{\frac{203}2}[/tex]
