GATE 2014 CS (CS3) – Question 16
If $\int_0^{2\pi}|x\sin x|\,dx=k\pi$, find k.
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Correct answer: 4 to 4
Explanation
Since x≥0, split where sinxchanges sign atπ. Integration by parts gives∫₀^πxsinx dx=πand−∫_π^(2π)xsinx dx=3π. The total is4π, hence **k=4**. Omitting the absolute value would instead give a negative integral.