Solution
= Solution
A <holomorphic primitive> on the slit domain is $P(z)=zL(z)-z$, since the <product rule> and $L'=1/z$ give $P'=L$. The <line segment> stays in that domain. The <fundamental theorem for line integrals> therefore gives
$$
\begin{aligned}
\int_{[1,i]}L(z)\,dz&=P(i)-P(1)\\
&=\left(i\frac{\pi i}{2}-i\right)-(-1).
\end{aligned}
$$
Hence
$$
\boxed{\int_{[1,i]}L(z)\,dz=1-\frac\pi2-i.}
$$
The value $L(i)=\pi i/2$ is fixed by the chosen branch.