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Question Number 126986 by mnjuly1970 last updated on 25/Dec/20
...nicecalculus...provethat::I:=∫0π2{cot(x)}cot(x)dx=12(π−ln(sinh(π)π)){x}isfractionalpartofx..
Answered by Olaf last updated on 26/Dec/20
I=∫0π2{cotx}cotxdxLetu=cotxI=∫∞0{u}u.du(−1−u2)I=∫0∞{u}u(1+u2)duI=∑∞n=0∫nn+1{u}u(1+u2)duI=∑∞n=0∫nn+1u−nu(1+u2)duI=∑∞n=0∫nn+1[nu+1u2+1−nu]duI=∑∞n=0[n2ln(1+u2)−arctanu−nlnu]nn+1...tobecontinued...
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