Evaluate the Integral ∫₀^∞ e⁻ˣ² dx and Show Its Value
The integral ∫₀^∞ e⁻ˣ² dx is a cornerstone example in calculus that illustrates how an apparently simple expression can be evaluated using clever techniques. Consider this: in this article we will evaluate the integral step by step, demonstrate why it converges, and reveal that its exact value is √π/2. Understanding this process not only sharpens analytical skills but also opens doors to applications in probability, physics, and engineering Not complicated — just consistent..
The Integral and Its Significance
The function e⁻ˣ² appears frequently in the Gaussian distribution, a fundamental model in statistics and natural sciences. When integrated over the entire positive real line, the result √π/2 provides the normalization constant for the half‑normal distribution. Also worth noting, this integral serves as a gateway to more advanced topics such as the Gamma function and complex analysis. Recognizing its importance helps students see why mastering its evaluation is worthwhile That alone is useful..
Preparing the Evaluation
Before diving into the computation, it is essential to treat the integral as an improper integral. An improper integral of the form ∫ₐ^∞ f(x) dx is defined as the limit:
∫ₐ^∞ f(x) dx = lim_{b→∞} ∫ₐ^b f(x) dx,
provided this limit exists and is finite. In our case, f(x) = e⁻ˣ², which decays rapidly as x → ∞, suggesting convergence. Near x =
