Integral Of Xe X From 0 To Infinity

The Definite Integral of xe⁻ˣ from 0 to Infinity: A Comprehensive Guide

The definite integral ∫₀^∞ xe⁻ˣ dx is a classic example in calculus, frequently encountered in probability, statistics, and various branches of physics and engineering. This integral doesn't yield to simple integration techniques, requiring a more sophisticated approach. This article will delve into the solution, exploring various methods and highlighting the significance of this integral. We'll also examine its practical applications and provide a step-by-step walkthrough to help you fully grasp the concept.

Understanding the Integral and its Significance

The expression ∫₀^∞ xe⁻ˣ dx represents the area under the curve of the function f(x) = xe⁻ˣ from x = 0 to x = ∞. This function is a decaying exponential function multiplied by a linear function (x). The exponential term ensures that the function approaches zero as x approaches infinity, making the integral convergent (meaning it has a finite value). This is crucial; otherwise, the integral would be divergent and lack a defined value.

This integral holds significant importance because:

• Probability and Statistics: It's fundamental in probability theory, particularly in the context of exponential distributions. The result of this integral often appears in calculations related to expected values and moments of exponential random variables. For instance, it's used to determine the mean of an exponential distribution.

• Physics and Engineering: Many physical phenomena are modeled using exponential decay, such as radioactive decay, the discharge of a capacitor, and certain chemical reactions. This integral frequently arises when calculating average values or other relevant quantities within these systems.

• Calculus and Mathematical Analysis: The integral serves as an excellent example to demonstrate the application of integration techniques like integration by parts and the concept of improper integrals.

Solving the Integral: A Step-by-Step Approach using Integration by Parts

The most common and effective method to solve this integral is integration by parts. This technique is based on the product rule for differentiation and allows us to integrate products of functions. The formula for integration by parts is:

∫u dv = uv - ∫v du

Where 'u' and 'v' are functions of x. To apply this to our integral, we make the following choices:

• u = x => du = dx
• dv = e⁻ˣ dx => v = -e⁻ˣ

Substituting these into the integration by parts formula, we get:

∫₀^∞ xe⁻ˣ dx = [-xe⁻ˣ]₀^∞ - ∫₀^∞ -e⁻ˣ dx

Let's break this down:

1. Evaluating [-xe⁻ˣ]₀^∞: This is a limit evaluation. As x approaches infinity, the term xe⁻ˣ approaches zero due to the exponential term dominating. As x approaches 0, the term approaches 0. Therefore:

   lim (x→∞) [-xe⁻ˣ] = 0

   lim (x→0) [-xe⁻ˣ] = 0

   So, [-xe⁻ˣ]₀^∞ = 0 - 0 = 0

2. Evaluating ∫₀^∞ e⁻ˣ dx: This is a straightforward integral:

   ∫₀^∞ e⁻ˣ dx = [-e⁻ˣ]₀^∞ = (- lim (x→∞) e⁻ˣ) - (-e⁰) = 0 - (-1) = 1

Putting it all together:

∫₀^∞ xe⁻ˣ dx = 0 - (-1) = 1

Therefore, the definite integral of xe⁻ˣ from 0 to infinity is 1.

Alternative Approaches and Verifications

While integration by parts is the most direct method, other approaches can be used to verify the result. These methods might be more complex or require more advanced mathematical knowledge but provide valuable insights into the problem.

• Gamma Function: The gamma function, denoted by Γ(z), is a generalization of the factorial function to complex numbers. It's defined as:

  Γ(z) = ∫₀^∞ t^(z-1)e⁻ᵗ dt

  If we set z = 2, we get:

  Γ(2) = ∫₀^∞ te⁻ᵗ dt

  Since Γ(2) = 1! = 1, this confirms our previous result.

• Laplace Transforms: Laplace transforms provide a powerful tool for solving differential equations and integrals. The Laplace transform of xe⁻ˣ is easily computed and its inverse transform gives the value of the integral. This method, however, requires familiarity with the theory of Laplace transforms.

• Numerical Integration: For those not comfortable with advanced calculus techniques, numerical integration methods like the trapezoidal rule or Simpson's rule can approximate the value of the definite integral. While these methods provide only approximate solutions, they can be valuable for verification or exploration in cases where analytical solutions are difficult to obtain. However, with the exponential decay, achieving high accuracy with numerical methods is usually straightforward.

Applications in Probability and Statistics: The Exponential Distribution

The result of this integral is deeply connected to the exponential distribution, a fundamental probability distribution used extensively in various fields. The probability density function (pdf) of an exponential distribution is given by:

f(x; λ) = λe⁻λx for x ≥ 0 and λ > 0

where λ is the rate parameter.

The expected value (or mean) of an exponential distribution is calculated as:

E[X] = ∫₀^∞ x * f(x; λ) dx = ∫₀^∞ xλe⁻λx dx

By substituting u = λx, du = λdx, the integral becomes:

E[X] = (1/λ) ∫₀^∞ ue⁻ᵘ du = (1/λ) * 1 = 1/λ

This highlights the direct relationship between the integral ∫₀^∞ xe⁻ˣ dx = 1 and the mean of an exponential distribution.

Conclusion: A Cornerstone Integral in Mathematics and its Applications

The integral ∫₀^∞ xe⁻ˣ dx = 1 is a cornerstone result in mathematics, appearing frequently in various fields. Its solution, primarily obtained through integration by parts, is fundamental to understanding probability distributions, solving differential equations, and modeling various real-world phenomena involving exponential decay. This integral’s simplicity belies its profound significance, serving as a testament to the power and elegance of mathematical analysis. Its numerous applications across diverse disciplines underscores the importance of mastering this key concept in calculus and its related fields. The detailed explanations and alternative methods provided in this article aim to equip readers with a comprehensive understanding, enabling them to tackle similar integrals and apply the knowledge to real-world problem-solving. Remember that consistent practice and a strong understanding of fundamental calculus concepts are essential for successfully navigating these types of mathematical challenges.