Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
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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