Integral Of Xe X 2 From 0 To Infinity

Evaluating the Definite Integral of xe^(x^2) from 0 to Infinity

The definite integral ∫₀^∞ xeˣ² dx presents a fascinating challenge in calculus. While seemingly straightforward, it requires a specific approach to solve. This article will delve into the detailed solution, exploring the underlying techniques and providing a comprehensive understanding of the process. We will also touch upon the broader applications of this type of integral in various fields.

Understanding the Problem: ∫₀^∞ xeˣ² dx

The integral ∫₀^∞ xeˣ² dx represents the area under the curve of the function f(x) = xeˣ² from x = 0 to x = ∞. This is an improper integral because the upper limit of integration is infinity. Improper integrals require careful consideration and often involve limit techniques. The integrand, xeˣ², is a product of a polynomial function (x) and an exponential function (eˣ²). This combination necessitates a specific integration strategy.

Solving the Integral Using Substitution

The most efficient method for solving this integral involves a u-substitution. Let's outline the steps:

1. Choosing the Substitution:

We strategically choose our substitution to simplify the integrand. Let:

u = x²

2. Finding the Differential:

Now we find the differential of u:

du = 2x dx

Notice that 'x dx' is present in our original integral. We can rearrange this to solve for 'x dx':

x dx = (1/2) du

3. Substituting into the Integral:

We substitute 'u' and 'du' into our original integral:

∫₀^∞ xeˣ² dx = ∫₀^∞ eᵘ (1/2) du

4. Simplifying the Integral:

The integral simplifies to:

(1/2) ∫₀^∞ eᵘ du

5. Evaluating the Integral:

The antiderivative of eᵘ is simply eᵘ. Thus, we have:

(1/2) [eᵘ]₀^∞

6. Applying the Limits of Integration:

Now, we substitute back x² for u:

(1/2) [eˣ²]₀^∞

This is where the improper integral nature becomes crucial. We evaluate the limit as x approaches infinity:

lim (x→∞) (1/2)eˣ² - (1/2)e⁰

7. Evaluating the Limit:

As x approaches infinity, eˣ² approaches infinity. Therefore, the limit is undefined in its current form. However, we must remember the context: we are dealing with a definite integral. The limit of (1/2)eˣ² as x approaches infinity is indeed infinity, which suggests the integral diverges. Let's examine this further.

Divergence of the Integral

The expression (1/2)[eˣ²]₀^∞ evaluates to:

(1/2) [lim (x→∞) eˣ² - e⁰]

The term lim (x→∞) eˣ² is unbounded and tends towards infinity. This leads us to the conclusion that the integral diverges.

This means the integral ∫₀^∞ xeˣ² dx does not have a finite value. The area under the curve extends infinitely.

Examining the Integrand and its Behavior

The function f(x) = xeˣ² grows rapidly as x increases. The exponential term, eˣ², dominates the linear term, x, resulting in an integrand that increases without bound. This rapid growth is the key reason why the integral diverges. Visualization of the graph of xeˣ² reinforces this understanding. The area under the curve from 0 to infinity is infinite.

Contrast with Convergent Integrals

It's important to contrast this divergent integral with examples of convergent improper integrals. For instance, integrals such as ∫₀^∞ e⁻ˣ dx converge to a finite value. The exponential decay in e⁻ˣ is crucial for its convergence. The difference highlights the importance of the integrand's behavior at infinity.

Applications and Significance

While this specific integral diverges, the techniques used to analyze it – u-substitution and evaluating limits of improper integrals – are fundamental to many areas of mathematics, science, and engineering. Similar types of integrals appear in:

• Probability and Statistics: Many probability density functions involve integrals similar in structure. Understanding how to evaluate these integrals is vital in calculating probabilities and expected values.

• Physics: Integrals of this form often appear in solving differential equations that describe physical phenomena, such as heat transfer or wave propagation.

• Engineering: Similar integrals arise in the analysis of various systems, from electrical circuits to mechanical structures.

• Financial Modeling: In finance, exponential functions are used to model growth and decay. Solving integrals similar to the one discussed here helps to assess long-term scenarios in investments and risk analysis.

Conclusion: Understanding Divergence and its Implications

The integral ∫₀^∞ xeˣ² dx diverges, meaning it doesn't have a finite value. This result stems from the rapid growth of the integrand xeˣ² as x approaches infinity. Understanding why this integral diverges underscores the importance of examining the behavior of integrands, especially when dealing with improper integrals. The techniques employed—u-substitution and the evaluation of limits—remain crucial tools for tackling a wide range of integration problems, providing insights into various scientific and engineering applications. While this specific integral yielded a divergent result, the process of attempting to solve it provides invaluable experience in applying fundamental calculus techniques and understanding the behavior of functions. The divergence itself is a significant piece of information, often indicating limitations or unbounded behavior within a system being modeled.