Improper Integrals: Definitions, Nature, and Properties

Authors: M. Aguiar, S. Furtado, J.M. Oliveira & H. Reis.
Institution: School of Economics and Management of the University of Porto (Faculdade de Economia do Porto - UP).
Subject Matter: The study of improper integrals of the first and second kinds, their definitions, convergence properties, and underlying theorems.

Conceptual Example: Consider the function $f(x) = e^{-x}$ .
Primary Objective: To calculate the area of the region in the 1st quadrant bounded by the horizontal axis (x-axis), the graph of the function $f$ , and the vertical line defined by $x = 0$ , provided this area is finite.
Visual Representation: This involves examining the behavior of the integral as the upper limit approaches $+\infty$ . * The area is calculated over an interval from $0$ to a variable point $t$ , and the limit is evaluated as $t \to +\infty$ .

Case 1: Infinite Upper Limit: * If $f = f(x)$ is a continuous function on the interval $[a, +\infty[$ , the improper integral is defined as: * $\int_{a}^{+\infty} f(x)\,dx = \lim_{t \to +\infty} \int_{a}^{t} f(x)\,dx$
Case 2: Infinite Lower Limit: * If $f = f(x)$ is a continuous function on the interval $]-\infty, b]$ , the improper integral is defined as: * $\int_{-\infty}^{b} f(x)\,dx = \lim_{t \to -\infty} \int_{t}^{b} f(x)\,dx$

Convergence: An improper integral of the 1st kind is said to be convergent if the corresponding limit exists and is a finite value.
Divergence: If the limit does not exist or is infinite, the improper integral is deemed divergent.
Value of the Integral: When the integral converges, the resulting finite limit is referred to as the "value" of the integral.

Relationship between Limits: A specific relationship exists between positive and negative infinite limits: * $\int_{-\infty}^{b} f(x)\,dx = \int_{-b}^{+\infty} f(-x)\,dx$
Step-by-Step Formal Proof: * 1. Start with the definition: $\int_{-\infty}^{b} f(x)\,dx = \lim_{t \to -\infty} \int_{t}^{b} f(x)\,dx$ * 2. Use the substitution $u = -x$ , which implies $du = -dx$ and changes limits from $[t, b]$ to $[-t, -b]$ : * $\lim_{t \to -\infty} \int_{-t}^{-b} -f(-u)\,du$ * 3. Swap the limits of integration to remove the negative sign: * $\lim_{t \to -\infty} \int_{-b}^{-t} f(-u)\,du$ * 4. As $t \to -\infty$ , the value $-t$ approaches $+\infty$ . Change the variable to $t'$ where $t' = -t$ : * $\lim_{t' \to +\infty} \int_{-b}^{t'} f(-x)\,dx = \int_{-b}^{+\infty} f(-x)\,dx$

Scalar Multiplication Theorem: * Let $f(x)$ be a continuous function in $[a, +\infty[$ and $\lambda \in \mathbb{R} \setminus \{0\}$ . * If $\int_{a}^{+\infty} f(x)\,dx$ converges and has the value $S$ , then $\int_{a}^{+\infty} \lambda f(x)\,dx$ also converges and its value is $\lambda S$ . * If $\int_{a}^{+\infty} f(x)\,dx$ diverges, then $\int_{a}^{+\infty} \lambda f(x)\,dx$ must also diverge.
Summation Theorem: * Let $f(x)$ and $g(x)$ be continuous functions in $[a, +\infty[$ . * Case of Mutual Convergence: If $\int_{a}^{+\infty} f(x)\,dx$ converges to $S_1$ and $\int_{a}^{+\infty} g(x)\,dx$ converges to $S_2$ , then $\int_{a}^{+\infty} (f(x) + g(x))\,dx$ converges and has the value $S_1 + S_2$ . * Case of Mixed Nature: If $\int_{a}^{+\infty} f(x)\,dx$ diverges and $\int_{a}^{+\infty} g(x)\,dx$ converges, then the integral of the sum $\int_{a}^{+\infty} (f(x) + g(x))\,dx$ is guaranteed to diverge.
Note on Mutual Divergence: The theorem does not provide a conclusion regarding the nature of $\int_{a}^{+\infty} (f(x) + g(x))\,dx$ when both $\int_{a}^{+\infty} f(x)\,dx$ and $\int_{a}^{+\infty} g(x)\,dx$ diverge. In this specific scenario, no conclusion can be drawn without further analysis.

Conceptual Example: Consider the function $f(x) = \frac{1}{x^{1/2}}$ .
Primary Objective: To calculate the area of the region bounded by the horizontal axis, the graph of $f$ , and the vertical lines $x = 0$ and $x = 1$ .
Characteristic Challenge: Unlike the 1st kind (where the interval was infinite), here the function itself becomes unbounded (infinite) at a point within or on the boundary of the interval of integration.
Visual Representation: In the example of $f(x) = \frac{1}{x^{1/2}}$ , the function approaches $+\infty$ as $x$ approaches $0$ from the right ( $x \to 0^{+}$ ).

Case 1: Unbounded at the Lower Limit: * If $f = f(x)$ is a continuous function on the interval $]a, b]$ and is not bounded as $x \to a^{+}$ , the integral is defined as: * $\int_{a}^{b} f(x)\,dx = \lim_{t \to a^{+}} \int_{t}^{b} f(x)\,dx$
Case 2: Unbounded at the Upper Limit: * If $f = f(x)$ is a continuous function on the interval $[a, b[$ and is not bounded as $x \to b^{-}$ , the integral is defined as: * $\int_{a}^{b} f(x)\,dx = \lim_{t \to b^{-}} \int_{a}^{t} f(x)\,dx$

Convergence: An improper integral of the 2nd kind is convergent if the relevant limits exist and are finite.
Divergence: If the limit does not exist or is infinite, the integral is divergent.
Value of the Integral: If the integral converges, the value resulting from the limit calculation is defined as the value of the integral.