I assume the integral is supposed to be
[tex]\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz[/tex]
Parameterize C by
[tex](x(t),y(t),z(t)) = (1-t)(1,0,0) + t(5,1,2) = (1+4t, t, 2t)[/tex]
with 0 ≤ t ≤ 1, and dx = 4 dt, dy = dt, and dz = 2 dt. Then the line integral is
[tex]\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = \int_0^1 (2t)^2(4\,dt) + (1+4t)^2\,dt + t^2(2\,dt)[/tex]
[tex]\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = \int_0^1(1+8t+34t^2)\,dt[/tex]
[tex]\displaystyle \int_C z^2\,dx + x^2\,dy + y^2\,dz = 1+4+\frac{34}3 = \boxed{\frac{49}3}[/tex]
