Integral over exponential involving reciprocial – math.stackexchange.com

I want to show $$I := \int_{-\infty}^\infty \exp \left(-\left(x-\frac p x \right)^2\right) \, dx = \sqrt{\pi}$$ for any non-negative $p\geq 0$. I tried to prove $I^2=\pi$ using Fubini's theorem, but ...

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