Topic 6: Improper Integrals

Improper Integrals Overview

• Definition: An improper integral involves either an infinite interval of integration or an unbounded function (e.g., a function that approaches infinity).

• Components Required: To compute improper integrals, we use the concepts of integration and limits.

Key Concepts and Notations

• Limits: Key notations include:-

  • Limit from the right: \lim_{x \to A^+} f(x)

  • Limit from the left: \lim_{x \to A^-} f(x)

• When both one-sided limits converge to the same value (L), we simply write: \lim_{x \to A} f(x) = L

Discontinuous Functions

• Piecewise Continuous Functions: These functions consist of separate, continuous pieces, each defined on an interval. Integrating piecewise continuous functions involves splitting the integral at points of discontinuity.

• Example: A function defined as: f(x) = \begin{cases} x + 1 & \text{if } x \leq 0 \ \frac{1}{2} x^2 & \text{if } x > 0 \end{cases}

  • Integration example from -1 to 2 is done by splitting it into two parts:

  • \int{-1}^{0} (x + 1) \, dx + \int{0}^{2} \left(\frac{1}{2} x^2\right) \, dx

Improper Integrals Characteristics

• Type 1 (Infinite Interval): Limit involves infinity, e.g., \int_{1}^{\infty} f(x) \, dx

• Type 2 (Function Unbounded): The function becomes infinite within the range of integration.

  • Example: \int_{0}^{2} \frac{1}{\sqrt{x}} \, dx where the function tends to infinity as x approaches 0.

Criteria for Identifying Improper Integrals

• Example 1: \int_{1}^{\infty} \frac{1}{x^2} \, dx (Improper due to infinity)

• Example 2: \int_{0}^{2} \frac{1}{x - 1} \, dx (Improper due to a vertical asymptote)

• Example 3: \int_{1}^{2} \frac{1}{x^2} \, dx (Proper, no issues with infinity or discontinuities).

Evaluating Improper Integrals

Example 19: Evaluating \int_{1}^{\infty} \frac{1}{x^2} \, dx

• Step 1: Replace with limit: \lim{L \to \infty} \int{1}^{L} \frac{1}{x^2} \, dx

• Step 2: Evaluate: = \lim{L \to \infty} \left(-\frac{1}{x}\right) \bigg|{1}^{L} = \lim_{L \to \infty} [0 + 1] = 1 (Converges)

Example 20: Evaluating \int_{1}^{\infty} \frac{1}{\sqrt{x}} \, dx

• Step 1: Same limit process: \lim{L \to \infty} \int{1}^{L} \frac{1}{\sqrt{x}} \, dx

• Step 2: Solve: = \lim{L \to \infty} \left(2\sqrt{x}\right) \bigg|{1}^{L} = \lim_{L \to \infty} [2\sqrt{L} - 2] = \infty (Diverges).

Integral Evaluations and Behavior

• Convergence vs Divergence: Convergence means the integral approaches a finite value; divergence indicates it approaches infinity.

  • Example: If \int_{1}^{\infty} \frac{1}{x} \, dx diverges (gives infinity).

Sum of Improper Integrals

• When evaluating \int{-\infty}^{\infty} f(x) \, dx, split at 0: \int{-\infty}^{0} f(x) \, dx + \int_{0}^{\infty} f(x) \, dx.

• Evaluating the areas separately provides clear insight into the visibility of convergence or divergence.

Example 24: Evaluating \int_{-\infty}^{\infty} \frac{1}{x^2 + 1} \, dx

• Split into two parts and evaluate separately using the antiderivative \tan^{-1}(x) and limits.

• Combine results to find the area under the graph is consistent for negative and positive parts (reflective symmetry).

Summary

• Improper integrals can either diverge or converge to finite values depending on the behavior of the function and limits involved. Methods of substitution and finding limits are essential for thorough analysis and results.

• Always be conscious of discontinuities and take care when dealing with infinite limits. Breaking integrals down into manageable segments helps to clarify behavior and conclusions.

• Key takeaway: Analyze the nature of the function and structure of the integral to classify and compute the integral accordingly.