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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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user 1591719
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See this. – Ragib Zaman Aug 05 '12 at 14:43
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@Ragib Zaman: thanks. Your solution is very nice. – user 1591719 Aug 05 '12 at 14:47
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It's a particular case of this. – Davide Giraudo Aug 05 '12 at 14:50
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I like to see there are so many approaches for each problem. :-) – user 1591719 Aug 05 '12 at 14:51
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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. – Joel Cohen Aug 05 '12 at 14:51
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@ Joel Cohen: I think you're right. – user 1591719 Aug 05 '12 at 14:54
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@Chris:Are you allowed to interchange differentiation with integration? Check Leibniz rule. – Mhenni Benghorbal Aug 05 '12 at 15:04
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@ Joel Cohen: right. That solution is already provided at the link given by Ragib Zaman. – user 1591719 Aug 05 '12 at 15:15
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@ Mhenni Benghorbal: I did it in a hurry and missed that critical issue around $0$. :D – user 1591719 Aug 05 '12 at 15:19