Answer
$$0$$
Work Step by Step
Our aim is to integrate the integral as follows:
$$\int^{\pi}_0 \int^{\pi}_0 \int^{\pi}_0 cos(u+v+w) \space du \space dv \space dw =\int^{\pi}_0 \int^{\pi}_0 [sin(w+v+\pi)-sin(w+v)] \space dv \space dw \\=\int^{\pi}_0 [(-cos(w+2\pi)+cos(w+\pi)]+(cos(w+\pi)-cos (w))] dw \\=[-sin(w+2\pi)+sin(w+\pi)-sin (w)+sin(w+\pi)]^{\pi}_0 \\=0$$
