Practicing Success
$\int\limits_0^\pi[\cot x] d x$, [.] denotes the greatest integer function, is equal to |
$\frac{\pi}{2}$ 1 -1 $-\frac{\pi}{2}$ |
$-\frac{\pi}{2}$ |
Let $I=\int\limits_0^\pi[\cot x] d x$ .....(i) $\Rightarrow I=\int\limits_0^\pi[\cot (\pi-x)] d x$ [Using : $\int\limits_0^a f(x) d x=\int\limits_0^a f(a-x) d x$] $\Rightarrow I=\int\limits_0^\pi[-\cot x] d x$ ....(ii) Adding (i) and (ii), we get $2 I=\int\limits_0^\pi\{[\cot x]+[-\cot x]\} d x$ $\Rightarrow 2 I=\int\limits_0^\pi-1 d x$ [∵ $[x]+[-x]=-1$, if $x \notin Z$] $\Rightarrow 2 I=-\pi \Rightarrow I=-\frac{\pi}{2}$ |