What is the value of the integral
\begin{align}
I(k) = \int\_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) + \cos(kx)}{\sin(x) + \cos(x)} \ dx
\end{align}
where $k$ is an integer ?

Is it possible to evaluate a related integral
\begin{align}
I(k) = \int\_{0}^{\frac{\pi}{2}} \frac{ \sin(kx) - \cos(kx)}{\sin(x) - \cos(x)} \ dx
\end{align}
where $k$ is an integer ?