JEEOlympiad
Evaluate I = \int_0^{\pi/2} x\,\cot x\,dx and give its value rounded…
Question
Evaluate I = \int_0^{\pi/2} x\,\cot x\,dx and give its value rounded to 4 decimal places.
✓ Verified answer: 1.0888checked by our engine — not a guess
Step-by-step solution
Integrate by parts with u=x, dv=\cot x\,dx so v=\ln\sin x: I = [x\ln\sin x]_0^{\pi/2} - \int_0^{\pi/2}\ln\sin x\,dx.
The boundary term vanishes: at \pi/2, \ln\sin(\pi/2)=0; at 0, x\ln\sin x \to 0.
So I = -\int_0^{\pi/2}\ln\sin x\,dx.
The classical value \int_0^{\pi/2}\ln\sin x\,dx = -\frac{\pi}{2}\ln 2 gives I = \frac{\pi}{2}\ln 2 = 1.570796\times0.693147 = 1.088793.
Final answer1.0888
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