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Fortunately, David Speyer's comment solves the mystery; \(G(t)\) falls out of doing the integration in Cartesian coordinates over a triangular region. Just for kicks, here's how I imagine an exercise based on this method would look like (this time for a multi-variable calculus class):

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Although this is simpler than the usual calculation of the Gaussian integral, for which careful reasoning is needed to justify the use of polar coordinates, it seems more like a certificate than an actual proof; you can convince yourself that the calculation is valid, but you gain no insight into the reasoning that led up to it.[1]

While reading Timothy Gowers's blog I stumbled on Scott Carnahan's comment describing an elegant calculation of the Gaussian integral \[ ∫_{-∞}^{∞} e^{-x^2} \, dx = \sqrt{π}\text{.} \] I was so struck by its elementary character that I imagined what it would be like written up, say, as an extra credit exercise in a single-variable calculus class:

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