$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$ Evaluate Integral

Evaluate

$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx$$

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Maple says $-\gamma/2$. –  AD. Oct 13 '12 at 12:30
@Chris'ssister: I like your question here. They are really like simple or difficult puzzles. Thanks for sharing them here. –  Babak S. Oct 13 '12 at 13:11
@Babak Sorouh: I'm really glad to read these words. Thank you! –  Chris's sis Oct 13 '12 at 13:14

Related problems: (I), (II). Recalling the Mellin transform of a function $f$

$$F(s) = \int_{0}^{\infty} x^{s-1}f(x) \,dx .$$

Then we consider the more general integral

$$F(s) = \int_{0}^{\infty} x^{s-1}\left(\cos x - e^{-x^2}\right) \, dx \,.$$

The value of the integral in our problem follows by taking the limit as $s\to 0$ in the above integral. Evaluating the above integral gives

$$F(s) = \Gamma \left( s \right) \cos \left( \frac{\pi \,s}{2} \right) - \frac{1}{2}\, \Gamma \left(\frac{s}{2} \right) \,.$$

Taking the limit as $s \to 0 \,,$ we get the desired result

$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \ dx = -\frac{\gamma}{2}\,.$$

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your way seems so simple! Thanks! (+1) –  Chris's sis Oct 13 '12 at 19:30
@Chris'ssister: You are welcome. –  Mhenni Benghorbal Oct 13 '12 at 19:43

The result is

$$\int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \, dx = -\frac{\gamma}{2},$$

where $\gamma$ is the Euler-Mascheroni constant.

Some direct calculations are available, but I prefer to consider it as a difference of some sort of log-singularities. you can find a slightly general method in this line of approach to calculate integrals of this form in my blog posting.

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thanks for the answer (+1) –  Chris's sis Oct 13 '12 at 12:35