Assuming that:
Definite Integral of e^(-x^2) dx over [0,infinity] = sqrt(pi)/2
Solve for
Definite Integral of e^(-ax^2) dx over [-infinity,infinity]
I don't know how to approach the new "a" term. I can't use u-substitution, integration by parts, partial fractions, or trig substitution. How do I do this?
Substitute x = t/sqrt(a). The integral then becomes:
a^(-1/2)Integral of e^(-t^2) dt over [-infinity,infinity] =
2 a^(-1/2)Integral of e^(-t^2) dt over [0,infinity] = sqrt(pi/a)
To solve the definite integral of e^(-ax^2) dx over the interval [-∞, ∞], we can use a technique called Gaussian integration, specifically the Gaussian Quadrature method. This method allows us to approximate the integral of a function without explicitly finding an antiderivative.
Gaussian Quadrature is based on the principle of approximating the integral of a function by evaluating it at specific sample points, called quadrature points, and then calculating their weighted sum.
The first step is to rewrite the integral in a standard form:
Definite Integral of e^(-ax^2) dx over [-∞, ∞] = 2 * Definite Integral of e^(-ax^2) dx over [0, ∞]
Next, we use a change of variables to transform the integral into a form that can be evaluated using Gaussian Quadrature. Let u = √a * x, and then substitute dx = du / √a in the integral:
2 * Definite Integral of e^(-ax^2) dx over [0, ∞] = 2 * ∫(e^(-a(u/√a)^2) / √a) du over [0, ∞]
= 2/√a * ∫(e^(-u^2)) du over [0, ∞]
Now, we can apply Gaussian Quadrature to approximate this integral.
Gaussian Quadrature works by choosing a set of quadrature points and corresponding weights. In this case, we can use the Gauss-Hermite quadrature, which is specifically designed for integrals involving the Gaussian function e^(-x^2).
The Gauss-Hermite quadrature approximates the integral as the sum of the function values at the quadrature points, multiplied by their respective weights:
2/√a * ∫(e^(-u^2)) du over [0, ∞] ≈ (√π/2) * Σ(e^(-u_i^2)) * w_i
where Σ represents summation, e^(-u_i^2) are the function values at the quadrature points u_i, and w_i are the corresponding weights.
The specific quadrature points and weights for Gauss-Hermite quadrature can be found in tables or computed using numerical methods.
By using the appropriate values from the Gauss-Hermite quadrature tables or calculating them numerically, you can approximate the value of the definite integral of e^(-ax^2) over the interval [-∞, ∞].
It's important to note that this method provides an approximation rather than an exact solution. The accuracy of the approximation depends on the number of quadrature points used and the quality of the quadrature rule employed.