Advanced Mathematics: Solving, I can't understand the answer, why I need to divide the area with x+y=0.5π, and I can't understand the calculation steps. I want to explain.

∵0≤x+y≤π/2, ? cos(x+y) ? = cos (x+y); When π/2≤x+y≤π, ? cos(x+y) ? =-cos(x+y), x+y=π/2 is the dividing point for removing "absolute number", and ∴ is divided by "x+y=π/2".

In the D 1 region shown in the figure, 0≤x≤π/2, 0 ≤ y ≤ π/2-x. In the D2 region shown in the figure, 0 ≤x≤π/ 2, π/2-x ≤ y ≤ π-x; π/ 2 ≤x≤π，0≤y≤π-x .

∫cos(x+y)dy=sin(x+y)+C, just take "x" as a "constant" to integrate y, and substitute it into the integral interval value to get the result of "definite integral".

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