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
∫∞0e−√xdx (since |x|=x when x>0)
Also, e−√x 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 e−√x< "some other function which converges" for x>1, thus ∫∞1e−√xdx converges.
In other words, I need a function which is always greater than e−√x and whose integral converges. I know that e−x and e−2x both converge, but these are both smaller than e−√x for x>1.
Tips would be appreciated. Thank you.
Answer
Hint: for x>75, ln(x2)<√x.
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