### 3.19. Calculate

$$
\int_{0}^{i} z \sin z d z
$$
S o l u t i o n. Since the function $f(z)=z \sin z$ is analytic in any domain $D \subset \mathbb{C}$, encompassing any contour $L$ connecting the points 0 and $i$ (the verification of this fact is left to the reader), by the corollary of Cauchy's theorem we have:

$$
\begin{gathered}
\int_{0}^{i} z \sin z d z=-\int_{0}^{i} z d \cos z=-\left.\{z \cos z\}\right|_{0} ^{i}+\int_{0}^{i} \cos z d z= \\
=-i \cos i+\left.\{\sin z\}\right|_{0} ^{i}=-i \cos i+\sin i=-i(\cos i+i \sin i)=-i e^{i \cdot i}=-\frac{i}{e}
\end{gathered}
$$