# How can I evaluate the integral of $\exp(-x^2-1/x^2)dx$ from 0 to infinity?

Question : integral of $\exp(-x^2-1/x^2)dx$ from $0$ to infinity

I think the answer is square root of $\pi/2 \cdot \exp(-2)$. If I change $-x^2-1/x^2$ to $-(x-1/x)^2-2$ then the above integral becomes $\exp(-2) \cdot$ integral of $\exp(-(x-1/x)^2)dx$ from $0$ to infinity

integral of $\exp(-x^2)dx$ from $0$ to infinity = square root of $\pi/2$, but then is the

integral of $\exp(-(x-1/x)^2)dx$ from $0$ to infinity=integral of $\exp(-x^2)dx$ from $0$ to infinity ?

-
@tony I think you are correct but the formatting is making it difficult to read! This might help you, meta.math.stackexchange.com/questions/107/… learning tex is easy! –  john w. Feb 8 '12 at 6:48
Thank you very much for your information!! –  Tony Feb 8 '12 at 11:39

The integral you want to evaluate is $\mathrm e^{-2}I$, where $I=\int\limits_0^{+\infty}\mathrm e^{-(x-1/x)^2}\mathrm dx$. Let us compute $I$. The change of variable $z=1/x$ yields $z\gt0$ and $\mathrm dz=z^2\mathrm dx$, hence $I=\int\limits_0^{+\infty}\mathrm e^{-(z-1/z)^2}\mathrm dz/z^2$. Summing these two expressions of $I$, one gets $2I=\int\limits_0^{+\infty}\left(1+1/x^2\right)\mathrm e^{-(x-1/x)^2}\mathrm dx$. The change of variable $u=x-1/x$ yields $u$ in the whole real line and $\mathrm du=\left(1+1/x^2\right)\mathrm dx$, hence $2I=\int\limits_{-\infty}^{+\infty}\mathrm e^{-u^2}\mathrm du$. Finally, this last integral is $\sqrt\pi$, hence $$\color{red}{\int\limits_0^{+\infty}\mathrm e^{-x^2-1/x^2}\mathrm dx=\frac{\sqrt\pi}{2\mathrm e^2}}.$$