Will you please help me figure out whether the following improper integrals converge or not?
∫∞0x22xdx
∫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+1ex≥1+1e=:C>0.
log(1+exex)x2≥logCx2
For (1), one can determine convergence again by comparing the integrand to a judiciously chosen function.
Additional hint For (1), show that x2≤ax 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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