Thursday, July 25, 2019

integration - Laplace transform of frac1t and resulting improper integral



Could someone please explain how
0exxdx diverges?
This is because the Laplace transform of 1t can be reduced to this integral which has to diverge. But the limit comparison test with ex shows that the integral converges.



Please help.




Thanks in advance.


Answer



Regarding the second part of the question, to add to carmichael561's answer,



To use the limit comparison test with ex, we need
exxex
throughout the interval over which integral takes place.



However, this is true only for x>1, and hence the test cannot be used.


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