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Question

Mathematics Question on integral

∫0π[cot x]dx,[??\int\limits^{\pi}_{{0}}[cot\,x]dx, [?? denotes the greatest integer function, is equal to

A

π2\frac{\pi}{2}

B

11

C

−1-1

D

−π2-\frac{\pi}{2}

Answer

−π2-\frac{\pi}{2}

Explanation

Solution

Let I=∫0π[cot x]dx ...(1)I=\int\limits^{\pi}_{{0}}[cot\,x]dx \,...(1) =∫0π[cot(π−x)]dx∫0π−−cot x]dx ...(1)=\int\limits^{\pi}_{{0}}[cot(\pi-x)]dx\int\limits^{\pi}_{{0}}--cot\,x]dx \,...(1) Adding (1) and (2) 2I=∫0π[cot x]dx+∫0π[cot x]dx=∫0π(−1)dx   2I=\int\limits^{\pi}_{{0}}[cot\,x]dx+\int\limits^{\pi}_{{0}}[cot\,x]dx=\int\limits^{\pi}_{{0}}(-1)dx\,\,\, [∵[x]+[−x]=−1 if x∉Z=0 if x∈Z]\left[\because\left[x\right]+\left[-x\right]=-1 \,if\,x\notin Z=0\,if\,x\in Z\right] =[−x]0π=−π=\left[-x\right]^{\pi}_{0}=-\pi ∴I=−π2\therefore I=-\frac{\pi}{2}