Examining the Convergence of the Improper Integral [math]int_{-infty}^{infty}x^2e^{-x^2}dx[/math]
Improper integrals are integrals that involve infinite limits of integration. Examining the convergence of an improper integral is an important step in understanding the behavior of the integral. In this article, we will discuss how to examine the convergence of the improper integral [math]int_{-infty}^{infty}x^2e^{-x^2}dx[/math].
Improper integrals involve infinite limits of integration.
The convergence of an improper integral can be examined by determining the behavior of the function.
The integral [math]int_{-infty}^{infty}x^2e^{-x^2}dx[/math] is convergent.
The value of the integral is 1.
An improper integral is an integral that involves infinite limits of integration. In the case of the integral [math]int_{-infty}^{infty}x^2e^{-x^2}dx[/math], the limits of integration are from negative infinity to positive infinity. This means that the integral is an infinite sum of terms, each of which is a function of x.
To examine the convergence of the integral, we must first determine the behavior of the function [math]x^2e^{-x^2}[/math]. This can be done by taking the derivative of the function and examining the sign of the derivative. If the derivative is positive, then the function is increasing, and if the derivative is negative, then the function is decreasing. In this case, the derivative of the function is [math]2xe^{-x^2}-2x^2e^{-x^2}[/math], which is negative for all values of x. This means that the function is decreasing, and thus the integral is convergent.
Once we have determined that the integral is convergent, we can then examine the value of the integral. To do this, we can use the substitution [math]u=x^2[/math], which yields the integral [math]int_{0}^{infty}ue^{-u}du[/math]. This integral can be evaluated using integration by parts, which yields the result [math]int_{0}^{infty}ue^{-u}du=1-e^{-u}[/math]. Substituting back for u yields the result [math]int_{-infty}^{infty}x^2e^{-x^2}dx=1[/math].
Good to know:
Improper Integral: An integral that involves infinite limits of integration.
Convergence: The process of approaching a limit.
Integration by Parts: A method of integration that involves the product rule.
In conclusion, we have examined the convergence of the improper integral [math]int_{-infty}^{infty}x^2e^{-x^2}dx[/math] and found that it is convergent. We have also evaluated the integral and found that its value is 1.
The information provided in this article is for educational purposes only and should not be used as a substitute for professional advice.
