Math Problem Statement

Evaluate the following integrals: 1. ∫₀⁺∞ x e^(-x²) dx, 2. ∫₋∞⁺∞ x e^(-x²) dx, 3. ∫₋∞⁺∞ x e^(x²) dx.

Solution

The image contains three integrals that need to be evaluated. Let's break down each of them and solve them one by one:

  1. Integral 1:

0+xex2dx\int_0^{+\infty} x e^{-x^2} \, dx

This integral is a standard Gaussian integral that can be solved using a substitution method.

  • Substitution: Let u=x2u = x^2, then du=2xdxdu = 2x \, dx, or xdx=12dux \, dx = \frac{1}{2} \, du.

The limits change accordingly:

  • When x=0x = 0, u=0u = 0,
  • When x=+x = +\infty, u=+u = +\infty.

Thus, the integral becomes:

0+xex2dx=120+eudu\int_0^{+\infty} x e^{-x^2} \, dx = \frac{1}{2} \int_0^{+\infty} e^{-u} \, du

=12[eu]0+=12(0(1))=12= \frac{1}{2} \left[ -e^{-u} \right]_0^{+\infty} = \frac{1}{2} \left( 0 - (-1) \right) = \frac{1}{2}

So, the result of the first integral is 12\frac{1}{2}.


  1. Integral 2:

+xex2dx\int_{-\infty}^{+\infty} x e^{-x^2} \, dx

This is a symmetric integral because the function xex2x e^{-x^2} is odd with respect to xx. The integral of an odd function over symmetric limits (from -\infty to ++\infty) is always zero. Thus, we have:

+xex2dx=0\int_{-\infty}^{+\infty} x e^{-x^2} \, dx = 0


  1. Integral 3:

+xex2dx\int_{-\infty}^{+\infty} x e^{x^2} \, dx

This integral is problematic because the exponential function ex2e^{x^2} grows very rapidly as x±x \to \pm \infty, and this makes the integral diverge. In other words, this integral does not converge to a finite value. Hence, the result is:

Divergent\text{Divergent}


Summary of Results:

  1. 0+xex2dx=12\int_0^{+\infty} x e^{-x^2} \, dx = \frac{1}{2}
  2. +xex2dx=0\int_{-\infty}^{+\infty} x e^{-x^2} \, dx = 0
  3. +xex2dx\int_{-\infty}^{+\infty} x e^{x^2} \, dx is divergent.

Would you like more details on any of these steps? Feel free to ask!

Related Questions:

  1. How can you solve integrals involving other Gaussian-like functions?
  2. What are the properties of odd and even functions in integrals?
  3. How do you handle divergent integrals in practice?
  4. What other techniques can be used for solving Gaussian integrals?
  5. How do substitution techniques help in simplifying integrals?

Tip:

For integrals involving Gaussian functions, substitution techniques and recognizing patterns like odd/even functions are crucial for simplifying the work.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Integration
Gaussian Functions
Symmetry of Functions

Formulas

Substitution Method
Gaussian Integral Formula
Odd Function Property in Integrals

Theorems

The Odd Function Integral Theorem

Suitable Grade Level

University Level (Calculus)