Integral evaluation problem on R

Question

I need to evaluate this integral but can't: $$\int_{-\infty}^{+\infty}e^{-\frac{1}{x^{2}}}\quad dx$$

Can someone help me please !

Answer 1

Hint: $\displaystyle\lim_{x\to\pm\infty}e^{-\frac1{x^2}}=1$.

Answer 2

For the indefinite integral, you get the error function

$$\int e^{-\frac{1}{x^{2}}} dx = x e^{-\frac{1}{x^{2}}} +\sqrt{\pi} \,\text{erf}\left(\frac1x \right) + C$$

where the first term will lead to divergence when integrating over the whole real line.

$e^{-\frac{1}{x^{2}}}$ is between $0$ and $1$, and subtracting $1$ before integrating leads to something finite

$$\int_{-\infty}^{+\infty}\left(e^{-\frac{1}{x^{2}}} -1\right) dx = -2 \sqrt{\pi} \approx -3.5449$$

again illustrating why you will get an infinite result without subtracting $1$