We need to evaluate the integral of e to the negative x squared from negative infinity to positive infinity. This is known as the Gaussian integral, one of the most important integrals in mathematics. The function e to the negative x squared creates a bell-shaped curve. Our goal is to find the exact area under this curve over the entire real line. We'll use an elegant approach involving polar coordinates.
To solve this integral, we use a clever trick. First, let I equal our integral. Then we consider I squared, which is the product of two identical integrals. We can write this as one integral with variable x times another with variable y. Since these variables are independent, we can combine them into a single double integral over the entire x-y plane. The integrand becomes e to the negative quantity x squared plus y squared.
Now we convert to polar coordinates. We substitute x equals r cosine theta and y equals r sine theta. This transforms our integrand from e to the negative x squared plus y squared into e to the negative r squared, since x squared plus y squared equals r squared. The differential area element dx dy becomes r dr d theta. The integration region, which was the entire x-y plane, becomes zero to infinity for r and zero to two pi for theta.
Now we evaluate the polar integral. We have the double integral from zero to two pi and zero to infinity of e to the negative r squared times r, dr d theta. First, we evaluate the inner integral with respect to r. Using the substitution u equals r squared, we get du equals two r dr. This transforms our integral into one half times the integral of e to the negative u from zero to infinity, which equals one half. Then we evaluate the outer integral: one half integrated from zero to two pi gives us one half times two pi, which equals pi. Therefore, I squared equals pi.
Finally, we take the square root to find our answer. Since I squared equals pi, and since the integrand e to the negative x squared is always positive, we take the positive square root to get I equals square root of pi. Therefore, the Gaussian integral from negative infinity to positive infinity of e to the negative x squared dx equals square root of pi, which is approximately 1.772. This beautiful result is fundamental in probability theory, quantum mechanics, and many other areas of mathematics and physics.