Can anyone solve this odd integral?

Question

Can anyone solve this odd integral?

$$\int\frac{e^{-50(\frac 1x-1)^2}}{x}\,dx$$

for $x>0$. I couldn't . . .

Answer 1

Integrating over $(0,1)$, and letting $t=\dfrac1x$ , followed by $u=t-1$, we are left with evaluating

$\displaystyle\int_0^\infty\frac{e^{-au^2}}{u+1}~du~=~\frac{\pi\cdot\text{erfi}\Big(\sqrt a\Big)-\text{Ei}(a)}{2~e^a}~$ for $a=50$, where erfi stands for the imaginary

error function, and Ei represents the exponential integral. $~$ See Liouville's theorem and the

Risch algorithm for more information.