Improper Integrals and Convergence in Calculus

Improper Integrals

Definition and Types

Improper integrals extend the concept of definite integrals to cases where the interval of integration is infinite or the integrand becomes unbounded within the interval. These integrals are essential in calculus for handling a broader class of functions and applications.

• Type I Improper Integrals: Integrals over infinite intervals, such as $\int_a^{\infty} f(x)\,dx$ or $\int_{-\infty}^b f(x)\,dx$.

• Type II Improper Integrals: Integrals where the integrand has a discontinuity (vertical asymptote) at a point in the interval, such as $\int_a^b f(x)\,dx$ where $f(x)$ is not defined at $a$, $b$, or some $c$ in $(a, b)$.

Key Point: To evaluate improper integrals, limits are used to replace the problematic endpoint or point of discontinuity.

Evaluating Improper Integrals

• Infinite Interval: $\int_a^{\infty} f(x)\,dx = \lim_{t \to \infty} \int_a^t f(x)\,dx$

• Discontinuous Integrand: If $f(x)$ is discontinuous at $b$, $\int_a^b f(x)\,dx = \lim_{t \to b^-} \int_a^t f(x)\,dx$

Example: Evaluate $\int_1^{\infty} \frac{1}{x^2}\,dx$

• Set up the limit: $\lim_{t \to \infty} \int_1^t \frac{1}{x^2}\,dx$

• Compute the antiderivative: $-\frac{1}{x}$

• Evaluate: $\lim_{t \to \infty} \left(-\frac{1}{t} + 1\right) = 1$

Convergence and Divergence

An improper integral converges if the corresponding limit exists and is finite. It diverges if the limit does not exist or is infinite.

• Convergent Example: $\int_1^{\infty} \frac{1}{x^2}\,dx$ converges to 1.

• Divergent Example: $\int_1^{\infty} \frac{1}{x}\,dx = \lim_{t \to \infty} \ln t - \ln 1 = \infty$ (diverges).

Comparison Test for Improper Integrals

The comparison test helps determine convergence or divergence by comparing the given function to another function whose behavior is known.

• If $0 \leq f(x) \leq g(x)$ for all $x \geq a$:

  • If $\int_a^{\infty} g(x)\,dx$ converges, then $\int_a^{\infty} f(x)\,dx$ also converges.

  • If $\int_a^{\infty} f(x)\,dx$ diverges, then $\int_a^{\infty} g(x)\,dx$ also diverges.

Example: Compare $\int_1^{\infty} \frac{1}{x^p}\,dx$ for different values of $p$.

• If $p > 1$, the integral converges.

• If $p \leq 1$, the integral diverges.

Absolute and Conditional Convergence

An improper integral is absolutely convergent if $\int_a^b |f(x)|\,dx$ converges. If $\int_a^b f(x)\,dx$ converges but $\int_a^b |f(x)|\,dx$ diverges, it is conditionally convergent.

Fundamental Theorem of Calculus (FTC) and Antiderivatives

Statement of the FTC

The Fundamental Theorem of Calculus connects differentiation and integration, providing a method to evaluate definite integrals using antiderivatives.

• Part 1: If $F(x)$ is an antiderivative of $f(x)$ on $[a, b]$, then $\frac{d}{dx} \int_a^x f(t)\,dt = f(x)$.

• Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F'(x) = f(x)$.

Example: $\int_0^1 2x\,dx = x^2 \Big|_0^1 = 1^2 - 0^2 = 1$

Antiderivatives and Notation

• Antiderivative: A function $F(x)$ such that $F'(x) = f(x)$.

• Notation: $\int f(x)\,dx = F(x) + C$, where $C$ is the constant of integration.

Table: Convergence of $\int_1^{\infty} \frac{1}{x^p}\,dx$

Value of $p$	Convergence	Explanation
$p > 1$	Converges	The integral approaches a finite value as $x \to \infty$.
$p = 1$	Diverges	The integral becomes $\ln x$, which increases without bound.
$p < 1$	Diverges	The antiderivative grows without bound as $x \to \infty$.

Key Terms and Concepts

• Improper Integral: An integral with an infinite limit of integration or an integrand with an infinite discontinuity.

• Convergence: The property of an integral to approach a finite value.

• Divergence: The property of an integral to increase without bound or fail to approach a finite value.

• Comparison Test: A method to determine convergence or divergence by comparing to a known function.

• Antiderivative: A function whose derivative is the given function.

• Fundamental Theorem of Calculus: The theorem linking differentiation and integration.

Additional info:

• Some content was inferred and expanded for clarity, including the structure of the comparison test and the table summarizing convergence for $\int_1^{\infty} \frac{1}{x^p}\,dx$.

• Examples and definitions were added to ensure the notes are self-contained and suitable for exam preparation.