Thursday, 7 August 2014

calculus - Proof that the improper integral frac1(x21) from 0 to infty is divergent.



01x21



It was said that this integral is divergent. I have tried splitting the integral. 101x21+11x21



Using this, I tried to prove 101x21 is divergent by comparison test, but there are no functions that lie under 1x21 that diverge (at least that I can think of).



I know the function diverges by Wolfram Alpha, but am struggling to prove it.


Answer




Notice that



1x21=1x+11x1131x1



for 1x2. Now compute



2tdxx1.


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