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$$\int |\sin(x) + \cos(x)|\ dx$$

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.

Tianlalu
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J. Lastin
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1 Answers1

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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.