Questions: When you integrate the identity ∫₀^∞ e^(-x²) dx = √π/2 The domain of x covers:

Solution Steps

The integral we are evaluating is

\[ \int_0^{\infty} e^{-x^2} \, dx \]

This integral is known to be half of the Gaussian integral over the entire real line.

The full Gaussian integral is given by

\[ \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi} \]

Calculating this gives us

\[ \sqrt{\pi} \approx 1.7725 \]

Since the integral from 0 to \( \infty \) is half of the full integral, we have

\[ \int_0^{\infty} e^{-x^2} \, dx = \frac{1}{2} \sqrt{\pi} \approx \frac{1.7725}{2} \approx 0.8862 \]

Final Answer