The value of _-( )/(2)^( )/(2)((1)/([x]+4))dx, where [ ] denotes the greatest integer func…

Question

The value of $\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{1}{[x]+4})dx$, where $[\cdot ]$ denotes the greatest integer function, is

Accepted Answer

Break at integer points: on $[-\pi/2,-1)$, $[x]=-2$; on $[-1,0)$, $[x]=-1$; on $[0,1)$, $[x]=0$; on $[1,\pi/2]$, $[x]=1$.

$$I = \frac12\left(\frac\pi2-1\right)+\frac13(1)+\frac14(1)+\frac15\left(\frac\pi2-1\right)$$

$$= \left(\frac\pi4-\frac12\right)+\frac13+\frac14+\left(\frac\pi{10}-\frac15\right) = \frac{7\pi}{20}-\frac{7}{60} = \frac{21\pi-7}{60} = \frac{7(3\pi-1)}{60}$$

Solutions · 1