Integral Xe X From 0 To Infinity

Evaluating the Definite Integral of xe⁻ˣ from 0 to Infinity

The definite integral ∫₀^∞ xe⁻ˣ dx represents a classic problem in calculus, frequently encountered in various fields like probability, statistics, and physics. This integral doesn't yield to elementary integration techniques directly, requiring a more sophisticated approach. This article will delve into the solution using integration by parts, exploring the underlying concepts and applications. We will also discuss the broader implications and significance of this integral.

Understanding the Integral and its Significance

The integral ∫₀^∞ xe⁻ˣ dx represents the area under the curve of the function f(x) = xe⁻ˣ from x = 0 to x = ∞. The function itself is a decaying exponential function multiplied by a linear function (x). This combination leads to a curve that starts at (0,0), increases to a maximum, and then asymptotically approaches zero as x approaches infinity.

The significance of this integral lies in its frequent appearance in:

• Probability Theory: This integral is crucial in probability distributions, particularly in the exponential distribution and the gamma distribution. It relates to the calculation of expected values and moments.
• Statistics: The result of this integral is foundational in various statistical analyses, especially when dealing with exponentially distributed data.
• Quantum Mechanics: Similar integrals appear extensively in quantum mechanics, related to wave functions and probability densities.
• Engineering: This integral often arises in the solutions of differential equations that model physical systems exhibiting exponential decay.

Solving the Integral using Integration by Parts

The integral ∫₀^∞ xe⁻ˣ dx cannot be solved directly using basic integration rules. Instead, we employ the technique of integration by parts. The formula for integration by parts is:

∫u dv = uv - ∫v du

We strategically choose our 'u' and 'dv' as follows:

• 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 examine each part separately:

Part 1: [-xe⁻ˣ]₀^∞

This part involves evaluating the expression -xe⁻ˣ at the limits of integration (0 and ∞).

• As x approaches infinity (∞), the term -xe⁻ˣ approaches 0. This is because the exponential term (e⁻ˣ) decays much faster than the linear term (x) grows. We can demonstrate this using L'Hôpital's rule:

  lim (x→∞) -xe⁻ˣ = lim (x→∞) -x/eˣ = lim (x→∞) -1/eˣ = 0

• When x = 0, the term -xe⁻ˣ evaluates to 0.

Therefore, [-xe⁻ˣ]₀^∞ = 0 - 0 = 0

Part 2: ∫₀^∞ e⁻ˣ dx

This is a straightforward integral:

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

Combining the Parts

Combining the results from Part 1 and Part 2, we obtain the final result:

∫₀^∞ xe⁻ˣ dx = 0 + 1 = 1

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

Verification and Further Exploration

The result of 1 might seem surprisingly simple, given the complexity of the initial integral. This highlights the power and elegance of integration by parts.

To further solidify our understanding, we can explore several avenues:

• Gamma Function: The integral ∫₀^∞ xⁿ⁻¹e⁻ˣ dx defines the Gamma function, denoted as Γ(n). Our integral is a specific case of the Gamma function where n = 2. The Gamma function has the property that Γ(n) = (n-1)! for positive integers n. Therefore, Γ(2) = (2-1)! = 1!, which confirms our result.

• Numerical Integration: We can use numerical methods, such as the trapezoidal rule or Simpson's rule, to approximate the integral. While these methods provide an approximate value, they should converge towards 1 as the number of subdivisions increases. This serves as a numerical verification of our analytical solution.

• Laplace Transforms: The Laplace transform provides another powerful method to evaluate this integral. The Laplace transform of tⁿ is given by n!/sⁿ⁺¹. Substituting n=1 we can verify the results.

Applications and Examples

The result of this integral has far-reaching applications across various disciplines. Here are a few illustrative examples:

1. Expected Value of an Exponential Distribution:

In probability theory, the exponential distribution is often used to model the time until an event occurs (e.g., the lifespan of a component, the time until a customer arrives). The probability density function (PDF) of an exponential distribution with parameter λ is given by:

f(x) = λe⁻λˣ, x ≥ 0

The expected value (or mean) of this distribution is given by:

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

By making a substitution (u = λx), this integral can be reduced to the form we solved earlier, leading to:

E[X] = 1/λ

2. Moments of a Probability Distribution:

Higher moments of a probability distribution can also be expressed as integrals. For instance, the second moment (related to variance) of the exponential distribution would involve an integral similar to the one we solved, but with x² instead of x.

3. Solving Differential Equations:

Differential equations involving exponential decay often appear in physics and engineering. The solution of these equations frequently involves integrals similar to the one we've explored.

4. Quantum Mechanics:

In quantum mechanics, the probability density of finding a particle at a certain position is often expressed as an integral involving exponential functions. Evaluating such integrals is crucial for calculating various quantum mechanical quantities.

Conclusion

The definite integral ∫₀^∞ xe⁻ˣ dx = 1 is a fundamental result with significant applications in various fields. Its solution, elegantly obtained through integration by parts, showcases the power of mathematical techniques in solving problems seemingly intractable at first glance. Understanding this integral and its associated concepts is essential for anyone working with probability, statistics, physics, or related disciplines. The result's simplicity belies its widespread importance and deep connections to several advanced mathematical concepts. The ability to solve this integral effectively highlights the importance of mastering techniques like integration by parts and understanding the behaviour of exponential functions in the context of calculus and beyond.