Parameterize the line segment [tex]\mathcal C[/tex] by
[tex]\mathbf r(t)=(x(t),y(t),z(t))[/tex]
[tex]\mathbf r(t)=(1-t)(0,0,0)+t(1,1,1)=(t,t,t)[/tex]
with [tex]0\le t\le1[/tex]. Now,
[tex]\displaystyle\int_{\mathcal C}(2z+4y)\,\mathrm dx+(4x-3z)\,\mathrm dy+(2x-3y)\,\mathrm dz[/tex]
[tex]\displaystyle=\int_{\mathcal C}(2z+4y,4x-3z,2x-3y)\cdot(\mathrm dx,\mathrm dy,\mathrm dz)[/tex]
[tex]=\displaystyle\int_{t=0}^{t=1}(6t,t,-t)\cdot(1,1,1)\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^16t\,\mathrm dt=3[/tex]
