Example 3. Compute the integral $\int_{C} e^{\bar{z}} d z$, where $C$ is the line segment $y=-x$, connecting the points $z_{1}=0$ and $z_{2}=\pi-i \pi$.
Solution. The parametric equations of the curve $C$ are

$$
x=t, \quad y=-t
$$

or in complex form

$$
z=t-i t
$$

where the real variable $t$ varies from 0 to $\pi$.

Applying formula (2), we get

$$
\int_{C} \mathrm{e}^{\bar{z}} d z=\int_{0}^{\pi} e^{t+i t}(1-i) d t=(1-i) \int_{0}^{\pi} e^{(1+i) t} d t=\left.\frac{1-i}{1+i} e^{(1+i) t}\right|_{0} ^{\pi}=\left(e^{\pi}+1\right) i
$$