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Best way to integrate $ \int_0^\infty \frac{e^{-at} - e^{-bt}}{t} \text{d}t $

Compute the improper integral: $$I=\int_{0}^{\infty} \frac{e^{-a x} - e^{-b x}}{x} dx$$

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    See [this](http://math.stackexchange.com/questions/62254/).2012-08-05
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    @Ragib Zaman: thanks. Your solution is very nice.2012-08-05
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    It's a particular case of [this](http://math.stackexchange.com/questions/61828/frullani-proof-integrals).2012-08-05
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    I like to see there are so many approaches for each problem. :-)2012-08-05
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    You can't chose $C_1$ and $C_2$ (their value is fixed), but they're equal because your equality $I(a) = \ln(a) + C_1$ is valid not for a single value, but for every $a > 0$ (in particular, you can plug in $b$). Also, your integral $I(a)$ is not actually defined (there's a problem around $0$). You can fix this by not splitting the integral, and differentiating.2012-08-05
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    @ Joel Cohen: I think you're right.2012-08-05
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    @Chris:Are you allowed to interchange differentiation with integration? Check Leibniz rule.2012-08-05
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    @ Joel Cohen: right. That solution is already provided at the link given by Ragib Zaman.2012-08-05
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    @ Mhenni Benghorbal: I did it in a hurry and missed that critical issue around $0$. :D2012-08-05

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