
The areas under three important curves from −∞ to +∞ equal specific constants.
For,
f(x) = e^{-x²}, ∫_{-∞}^{+∞} e^{-x²} dx = √π. For f(x) = 1/(x² + 1), ∫_{-∞}^{+∞} 1/(x² + 1) dx = π.
For f(x) = sin(x)/x, ∫_{-∞}^{+∞} sin(x)/x dx = π.
These constants are used to define normalized probability distributions in statistics and to analyze frequency content in signal processing applications.

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