How should I calculate $\int_0^\infty e^{-\frac{1}{2}(x^2+a^2/x^2)}\,dx$

Question

$$\int_0^\infty e^{-\frac{1}{2}(x^2+a^2/x^2)}\,dx$$

I even know the answer but indefinite integral is much more complicated than answer. The answer, according to Wolfram is $\sqrt{\pi/2}\,e^{-a}$.

Answer 1

Let $$I=\int_0^\infty e^{-\frac{1}{2}(x^2+a^2/x^2)}\,dx. $$ Then under $x\to a^2x$, one has $$I=\int_0^\infty a^2e^{-\frac{1}{2}a^2(x^2+1/x^2)}\,dx=\int_0^\infty e^{-\frac{1}{2}a^2(x-1/x)^2}\,dx$$ Under $x-\frac1x \to x$, one can get the answer.