Thursday, 15 January 2015

calculus - Does intinftyinftyesqrt|x|,mathrmdx converge?




I would like to find out if this integral converges: e|x|dx




Since this is a symmetric function I figured I could focus on only one side of the integral, namely




0e|x|dx which in this case is equivalent to
0exdx (since |x|=x when x>0)



Also, ex is bounded from 0 to 1 meaning the integral there is a constant, so I will use the integral from 1 to .



I know this converges (checked with a calculator) but cannot seem to find an argument for the comparison test to say that since ex< "some other function which converges" for x>1, thus 1exdx converges.



In other words, I need a function which is always greater than ex and whose integral converges. I know that ex and e2x both converge, but these are both smaller than ex for x>1.



Tips would be appreciated. Thank you.



Answer



Hint: for x>75, ln(x2)<x.


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