I am trying to compute this integral:
$\int_{1}^2 \frac{e^{-x^2}}{x} dx$
But I'm confused on how to do this since I'm aware that $e^{-x^2}$ has no integrand.
I am trying to compute this integral:
$\int_{1}^2 \frac{e^{-x^2}}{x} dx$
But I'm confused on how to do this since I'm aware that $e^{-x^2}$ has no integrand.
This integral is not elementary. Change variables. Let $z=x^2$. Then $\begin{eqnarray} I &=& \int_1^2 dx\, x^{-1} e^{-x^2} \\ &=& \frac{1}{2} \int_1^4 dx\, z^{-1} e^{-z} \\ &=& \frac{1}{2} \left(\int_1^\infty dx\, z^{-1} e^{-z} - \int_4^{\infty} dx\, z^{-1} e^{-z}\right) \\ &=& \frac{1}{2}[\mathrm{E}_1(1) - \mathrm{E}_1(4)] \\ &=& 0.10780\cdots \end{eqnarray}$ where the integral $\mathrm{E}_1(x)$ is closely related to the exponential integral $\mathrm{Ei}(x)$. (This can in fact be rewritten as $\frac{1}{2}\left[\mathrm{Ei}(-4) - \mathrm{Ei}(-1)\right]$.) An integral such as this cannot be written in terms of simple functions---this is as simple as it gets!