$$\int_C z e^ {-z} dz$$ with $C$ being any curve from $i$ to $1 + i$
I defined $C$ as $$z(t) = t + 1 $$ $$i \leq t \leq 1 + i $$
Such that
$$\int_C f(z)dz = \int_{i}^{1+i} (t + 1)e^{-(t+1)}dt $$ $$=e^{-2-i}-e^{-1-i}$$
Does that seem correct? Is the definition of C ok or it could be more general?
Your parametrization seems to be wrong as you took a line between $\;1+i\;$ and $\;2+i\;$ , which is not what is given.
My method is as follows: $\;ze^{-z}\;$ has an easy primitive function, which is $\; -e^{-z}(z+1)\;$ (in real calculus two this means the function has a potential or that it is a conservative field), and then the result of the line integral is the difference of the primitive at the end point minus its value at the beginning, and no matter what path you choose to take:
$$\left.\int_Cze^{-z}dz=-e^{-z}(z+1)\right|_i^{1+i}=(1+i)e^{-i}-(2+i)e^{1+i}$$
