The value of  $\underset{-20\pi }{\overset{20\pi }{\int }}[\sin x+\cos x]dx$  is: (Where  $[.]$ denotes greatest integer function)

$$\begin{gathered} [\sin x+\cos ]=\left[\sqrt{2}\sin (x+\frac{\pi }{4})\right]=\{\begin{array}{l}10\le x\le \frac{\pi }{2}\\ 0\frac{\pi }{2}\le x\le \frac{3\pi }{4}\\ -1\frac{3\pi }{4}<x<\pi \\ -2\pi <x\le \frac{3\pi }{2}\\ -1\frac{3\pi }{2}<x\le \frac{7\pi }{2}\\ 0\frac{7\pi }{4}<x\le 2\pi \end{array} \\ I=20[\underset{0}{\overset{\pi /2}{\int }}1dx+\underset{\pi /2}{\overset{3\pi /4}{\int }}0dx+\underset{3\pi /4}{\overset{\pi }{\int }}-1dx+\underset{\pi }{\overset{3\pi /2}{\int }}-2dx+\underset{3\pi /2}{\overset{7\pi /4}{\int }}-1dx+\underset{7\pi /4}{\overset{2\pi }{\int }}0dx] \\ =20\times [\frac{\pi }{2}-\pi +\frac{3\pi }{4}-2\times \frac{3\pi }{2}+2\pi -\frac{7\pi }{4}+\frac{3\pi }{2}]=20\times -\pi =-20\pi \end{gathered}$$