Evaluate $\displaystyle I(a)=\int_0^{\infty} e^{-a(x^2 + x^{-2})} \ dx, \ \ 0 < a \in \mathbb{R}.$
i'm quite rusty, it's been a long time i don't do many integration problems.
it's $\displaystyle \int_{0}^{\infty }{e^{-a\left( x^{2}+x^{-2} \right)}\,dx}=\int_{0}^{1}{e^{-a\left( x^{2}+x^{-2} \right)}}+\int_{1}^{\infty }{e^{-a\left( x^{2}+x^{-2} \right)}},$ thus $\displaystyle \int_{1}^{\infty }{\left( 1+\frac{1}{x^{2}} \right)e^{-a\left( x^{2}+x^{-2} \right)}\,dx}=\int_{0}^{\infty }{e^{-a\left( x^{2}+2 \right)}\,dx},$ and your integral equals $\displaystyle \frac{1}{2e^{2a}}\sqrt{\frac{\pi }{a}}.$