Friday, 3 June 2016

calculus - Prove that inti0nftyfrac1tx+1dt=fracpixcscfracpix

I'm stuck on this identity:



01tx+1dt=πxcscπx



Could someone show me a proof for this?



What I've tried:




I've thrown a bunch of substitutions and integration by parts at this, but they haven't led me to the answer. I did, however, find these identites:
01tx+1dt=0tx2tx+1dt=10(1tt)1/xdt
But none of these seem to lead anywhere helpful.



I also tried introducing another variable to turn it into something similar to the Laplace Transform, but I'm not very familiar with methods like that, so they've led nowhere.

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