Evaluate the following integral from 0 to infinity

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Evaluate the following integral from 0 to infinity. (see attached for better picture)

e^(-ax)-e^(bx)
------------------ dx
x

Remarks:

a , b > 0
a < b
 

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Have you tried residue calculus?
 


No, we have not learned how to calculate integrals with residue calculus just yet.

Do you think it's possible with integration by parts?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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