$$\int |\sin(x) + \cos(x)|\ dx$$
Do I just do: $$\operatorname{sgn}(\sin(x) + \cos (x)) \int \sin(x) + \cos(x) \ dx = \frac{\sin(x) + \cos (x)}{|\sin(x) + \cos(x)|} \int \sin(x) + \cos(x)\ dx$$
and then continue normally? The abs. value here is throwing me for a loop.
Just write $\sin(x)+\cos(x)$ as ${\sqrt{2}}\sin(x+\frac{\pi}{4})$. Now you can split the domain into parts where $\sin(x+\frac{\pi}{4})$ is positive and negative, and integrate accordingly.