I wish to compute the following line integral $$\int_{C}(x-iy)dz$$ where $z(t)=(e^t- 1,t)$, $t \in [0,2]$
$dz = dx + idy = (e^t - 1 + i)dt$ We then have $\int_{C}(e^{t} - 1 - it)(e^t - 1 + i)dt$
I'm confused on what information we have about our contour $C$. Can anyone explain this?
$$\int_0^2(e^{t} - 1 - it)(e^t - 1 + i)\,dt$$
$$=\int_0^2(e^t-1)^2+t+i[e^t-1-te^t+t]\,dt$$
$$=\left[\frac{e^{2t}}{2}-2e^t+t+\frac{t^2}{2}+i\{e^t-t-te^t+t+\frac{t^2}{2}\}\right]_0^2$$