Convergence Test for Improper Integrals

Описание: This lecture discusses methods for determining the convergence or divergence of improper integrals.
The lecture covers:
It starts by reviewing two basic examples:
$\int_{a}^{\infty} x^{-n} dx$ (converges if n greater than 1, diverges if n ≤ 1).
$\int_{0}^{a} x^{-n} dx$ (converges if n less than 1, diverges if n ≥ 1).
If f(x) and g(x) are positive, integrable, and f(x) ≤ g(x) for x ≥ a, then ∫ f(x) dx converges if ∫ g(x) dx converges.
If f(x) ≥ g(x) for x ≥ a and ∫ g(x) dx diverges, then ∫ f(x) dx also diverges.
Several examples are provided to illustrate this test.
If x^μ f(x) is bounded for x ≥ a, then ∫ f(x) dx is absolutely convergent, provided μ greater than 1.
In particular, if lim (x→∞) x^μ f(x) exists (where μ greater than 1), then ∫ f(x) dx converges absolutely.
If lim (x→∞) x^μ f(x) exists and is not equal to zero (where μ ≤ 1), then ∫ f(x) dx diverges.
If ∫ f(x) dx converges absolutely and g(x) is bounded and integrable, then ∫ f(x)g(x) dx is absolutely convergent. This test is noted as "not very useful."
If f(x) and g(x) are positive and f(x) ≤ g(x) in (a+ε, b), then ∫ f(x) dx converges if ∫ g(x) dx converges.
If (x-a)^μ f(x) is bounded in (a,b), then ∫ f(x) dx is absolutely convergent, provided μ less than 1. It diverges if μ ≥ 1.
In particular, if lim (x→a+0) (x-a)^μ f(x) exists (where μ less than 1), then ∫ f(x) dx converges absolutely.
If lim (x→a+0) (x-a)^μ f(x) exists and is not zero (where μ ≥ 1), then ∫ f(x) dx diverges.
The lecture provides detailed discussions on the convergence and divergence of various integrals, including:
∫_0^1 x^(n-1) log(x) dx (analyzed for n=1, ngreater than1, and n less than1).
∫_0^∞ x^(α-1) e^(-x) dx (Gamma function, shown to converge for α greater than 0).
∫_0^1 x^(m-1) (1-x)^(n-1) dx (Beta function, shown to converge for m greater than 0, n greater than 0).
∫_1^∞ (log x) / x^p dx (evaluated for p greater than 1 using integration by parts).
∫_0^∞ e^(-x) cos(x) dx (evaluated using integration by parts).
∫_0^∞ e^(-√x) dx (evaluated by substitution).