I'm having trouble understanding the concept behind complex line integrals. I think they follow the same general steps and line integrals in the real $x,y$ plane, but I'm not sure.
$$\int_C z^2dz$$ $$C: z(t)=t^2+3t$$ $$0 <t <2$$
My attempt: $$dz=2t+3$$ $$z^2=t^4+6t^3+9t^2$$ $$\int_C z^2dz=\int_0^2 (t^4+6t^3+9t^2)(2t+3)dt$$ $$=\frac{1358}{3}$$
But I mainly solved this using half intuition and half guessing. I'm not confident in my understanding of this concept.