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$Evaluate \int_{0}^{π / 2} x \cot x d x$

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Correct option is A

Integrating by parts, taking cot⁡x as second function, given integral becomes I=[xlog⁡sin⁡x]0π/2−∫0π/2 log⁡sin⁡xdx=0−limx→0 (xlog⁡sin⁡x)−∫0π/2 log⁡sin⁡xdx=12πlog⁡2 as limx→0 xlog⁡sin⁡x=limx→0 log⁡sin⁡x1/x=limx→0 cot⁡x−1/x2 (Using L'Hopital Rule) =limx→0 −x2tan⁡x=limx→0 −x×xtan⁡x=0×1=0

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