Wednesday, January 23, 2019

calculus - Determining convergence of improper integrals including int10fraclnleft(1+exright)xx2textdx



Will you please help me figure out whether the following improper integrals converge or not?





  1. 0x22xdx


  2. 10ln(1+ex)xx2dx





As for the first one, I have no idea.
As for the second,
I have tried rewriting it as:

10ln(1+exex)x2dx


but I have no idea if it helps me or not.



Thanks in advance.


Answer



Hint You're on the right track with your manipulation in (2). Since ex is increasing, for all x[0,1] we have 1+exex=1+1ex1+1e=:C>0.

Since log is increasing, we have
log(1+exex)x2logCx2
on (0,1]. What can you say about the integral 10logCx2dx
relevant to the (direct) comparison test?




For (1), one can determine convergence again by comparing the integrand to a judiciously chosen function.




Additional hint For (1), show that x2ax for all sufficiently large x for some suitable constant a.




Remark One can also determine the convergence of the integral in (1) by evaluating it directly, though this is probably slower: Applying integration by parts twice shows that it has value 2(log2)3.


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