Friday, 22 April 2016

analysis - Improper integral convergence: int10fracmidlog(x)midasqrt1x2dx



As its told on the title I want to check the convergence/divergence of the improper integral when aR: 10log(x)a1x2dx
So, it's improper at x=0 and x=1, so I split the integral in :
120log(x)a1x2dx+112log(x)a1x2dx

I see, that on the first one the integrals it's like log(x)adx by the comparison limit test, but I don't know how to prove that log(x)adx converges.



Hopefully you can help me. Much thanks!


Answer



Note that



120log(x)a1x2dx



converges a by comparison test with 1x.




For the second part



112log(x)a1x2dx



let 1x2=y2



032log(1y2ay(y1y2)dy=320log(1y2a1y2dy




and note that for y0



log(1y2)∣∼y22



thus the integral converges for 2a<1 that is a>12 and diverges for 2a1 that is a12 by comparison with y2a.


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