Evaluating Integral involving exp: $ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$

Question

I am stuck at the following integral :-

$$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$

Can anybody give me some hint.

and also for this function

$$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-(x-u)^2-\frac{1}{(x-q)^2}\right)\,dx$$

where $u$ and $q$ are constants.

Thanks

Answer 1

Notice how the function is odd. Therefore, $\displaystyle \int_{-\infty}^{0} f(x)=-\int_{0}^{\infty} f(x)\implies \int_{-\infty}^{\infty}f(x)=0$ If we can show that $\displaystyle \lim\limits_{x\to0,\pm\infty}f(x)$ exists, which are where it can diverge, then the integral exists. As we can plainly see $\lim_{x\to0 \text{or} \pm\infty}$ is $0$. Therefore the integral converges to $0$.

Answer 2

One hint that might help is to note that the function you are integrating is an odd function, so ...