_0^ [ x] d x, [.] denotes the greatest integer function, is equal… - Vidyadip

MCQ

$\int_0^\pi[\cot x] d x$, [.] denotes the greatest integer function, is equal to

A
$\frac{\pi}{2}$
B
1
C
-1
✓
$-\frac{\pi}{2}$

✓

Answer

Correct option: D.

$-\frac{\pi}{2}$

(D)
Let $I =\int_0^\pi[\cot x] d x$ ...(i)
$\Rightarrow I =\int_0^\pi[\cot (\pi-x)] d x$
$\ldots\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$\Rightarrow I=\int_0^\pi[-\cot x] d x$ ...(ii)
Adding (i) and (ii), we get
$2 I =\int_0^\pi\{[\cot x]+[-\cot x]\} d x$
$\Rightarrow 2 I =\int_0^\pi-1 d x \ldots .[\because[x]+[-x]=-1$, if $x \notin Z ]$
$\Rightarrow 2 I =-\pi$
$\Rightarrow I=-\frac{\pi}{2}$

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