Already one of the Bernoullis, I think Daniel, asked Euler what number theory was good for. He replied that solving diophantine equations such as $x^2 + y^2 = 1$ and similar ones via rational parametrizations would allow to transform integrals $\int dx/\sqrt{1-x^2} $ into rational ones. Perhaps not that exciting today, but it's a time-honoured answer.
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