Chapter 17 — Key Takeaways

A structured recap of Improper Integrals. The single organizing question of the whole chapter: does the defining limit exist as a finite number? If yes, the integral converges; if no, it diverges.

The Two Types of Improper Integral (Sections 17.1, 17.2)

An improper integral is an ordinary definite integral with a limit wrapped around it. There are two ways an integral can be improper — and they can occur together.

Type 1 — Infinite limit of integration (Section 17.1). The interval is unbounded:

$$\int_a^\infty f(x)\,dx := \lim_{t\to\infty}\int_a^t f(x)\,dx, \qquad \int_{-\infty}^b f(x)\,dx := \lim_{s\to-\infty}\int_s^b f(x)\,dx.$$

For a doubly infinite interval, split at any convenient point $c$ and require both halves to converge separately: $$\int_{-\infty}^\infty f = \int_{-\infty}^c f + \int_c^\infty f.$$ You may not let $+\infty$ on one side cancel $-\infty$ on the other.

Type 2 — Infinite integrand (Section 17.2). The interval is finite but $f$ blows up at an endpoint:

$$\int_a^b f(x)\,dx := \lim_{t\to b^-}\int_a^t f(x)\,dx \quad(\text{singularity at } b),$$

and symmetrically at $a$. For an interior singularity at $c\in(a,b)$, split at $c$ and again require each one-sided piece to converge on its own.

The p-Integral Rules — Memorize Both (Sections 17.1, 17.2)

The two most-used convergence tests in practice are mirror images across the line $p = 1$:

$$\int_1^\infty \frac{dx}{x^p}\ \text{converges} \iff p > 1 \qquad(\text{need *fast decay* at infinity}),$$

$$\int_0^1 \frac{dx}{x^p}\ \text{converges} \iff p < 1 \qquad(\text{need a *mild blow-up* at zero}).$$

The exponent $p = 1$ — the function $1/x$ — fails on both ends, because its antiderivative is the logarithm, which grows and decays just slowly enough to straddle the borderline. The single most common student error is applying the wrong inequality; the directions are opposite.

Comparison Tests — Deciding Without Evaluating (Section 17.3)

When no elementary antiderivative exists, compare to an integrand you understand. Both tests require the functions to be non-negative (at least eventually).

Direct comparison. If $0 \le f \le g$: - $\int g$ converges $\Rightarrow$ $\int f$ converges (smaller area trapped beneath finite area). - $\int f$ diverges $\Rightarrow$ $\int g$ diverges.

Limit comparison. If $f,g > 0$ and $\displaystyle\lim_{x\to\infty}\frac{f(x)}{g(x)} = c$ with $0 < c < \infty$, then $\int f$ and $\int g$ share the same fate. Pick $g = 1/x^p$ by keeping only the leading powers of the numerator and denominator, then confirm with the limit.

If the integrand changes sign, the tests do not apply directly — test $|f|$ instead (see absolute convergence below).

The Gamma Function (Section 17.4)

$$\Gamma(s) = \int_0^\infty x^{s-1}e^{-x}\,dx, \qquad s > 0.$$

This improper integral converges for every $s > 0$: $e^{-x}$ crushes the tail at infinity, and near zero $x^{s-1}$ is an integrable singularity exactly when $s > 0$. Its three defining properties:

  • Recursion: $\Gamma(s+1) = s\,\Gamma(s)$ (proved by parts).
  • Base case: $\Gamma(1) = 1$.
  • Factorial: $\Gamma(n) = (n-1)!$ for positive integers $n$.
  • Half-integer: $\Gamma(1/2) = \sqrt{\pi}$ (via the substitution $x = u^2$ linking it to the Gaussian integral).

$\Gamma$ is the factorial smoothed into a continuous function; it supplies the normalizing constants of the Gamma, chi-squared, and Beta distributions and appears throughout special-function theory.

The Gaussian Integral (Section 17.5)

$$\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}.$$

A beautiful tension: $e^{-x^2}$ has no elementary antiderivative (Liouville's theorem; the antiderivative is named $\operatorname{erf}$), yet the definite integral over $\mathbb{R}$ is the exact value $\sqrt{\pi}$. The trick is to square the integral, read it as a double integral over the plane, and switch to polar coordinates where the radial symmetry makes it elementary — the full proof waits for Chapter 32. This integral is the source of the $\sqrt{2\pi}$ that normalizes the standard normal density $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$.

Subtleties: Absolute vs. Conditional Convergence (Section 17.7)

  • Absolute convergence: $\int f$ converges absolutely if $\int |f|$ converges. This is the robust kind — it survives comparison tests and numerical evaluation.
  • Conditional convergence: an integral can converge through cancellation of signed areas even when $\int|f| = \infty$. The classic case is the Dirichlet integral $\int_0^\infty \frac{\sin x}{x}\,dx = \frac{\pi}{2}$, whose absolute version diverges.
  • Cauchy principal value: a symmetric limit, e.g. $\operatorname{PV}\!\int_{-\infty}^\infty f = \lim_{T\to\infty}\int_{-T}^T f$, can be finite even when the true integral diverges. It is a different object — always keep the "PV" label.

Common Errors to Avoid

  • Blind FTC across an interior singularity. $\int_{-1}^1 \frac{dx}{x^2} \ne -2$; the integrand explodes at $x=0$, so the integral diverges. Always scan for singularities inside the interval before reaching for an antiderivative.
  • Using the wrong p-rule. At infinity you need $p > 1$; at zero you need $p < 1$. Opposite directions.
  • Letting infinities cancel. Both halves of a doubly infinite integral (or a split interior singularity) must converge separately. Symmetric cancellation gives only the principal value, not the integral.
  • Applying comparison tests to sign-changing integrands. Test $|f|$ first.

Applications Seen This Chapter (Section 17.6)

  • Pharmacokinetics (biology): total drug exposure $\text{AUC} = \int_0^\infty C_0 e^{-kt}\,dt = C_0/k$.
  • Probability / data science (Section 17.5): density normalization $\int f = 1$ defines every distribution's constant.
  • Physics: escape-velocity work $\int_{r_0}^\infty \frac{GMm}{r^2}\,dr = GMm/r_0$ (a $p=2$ integral).
  • Economics: present value of a perpetuity $\int_0^\infty R e^{-rt}\,dt = R/r$.
  • Engineering: the Laplace transform $\int_0^\infty e^{-st}f(t)\,dt$ and Fourier transform are themselves improper integrals.

Connections (Section 17.8)

Every convergence idea here returns as the integral test for series in Chapter 22: if $f$ is positive, continuous, and decreasing on $[1,\infty)$, then $\sum_{n=1}^\infty f(n)$ and $\int_1^\infty f(x)\,dx$ converge or diverge together. In particular the p-integral rule becomes the p-series rule ($\sum 1/n^p$ converges $\iff p > 1$), and absolute vs. conditional convergence has an exact series analog (the Dirichlet integral mirrors the alternating harmonic series). Improper integration is the dress rehearsal; series convergence is opening night.

Skills You Should Now Have

  • Recognize when an integral is improper (infinite limit, infinite integrand, or both).
  • Set up and evaluate convergent improper integrals via the limit definition.
  • Apply both p-integral rules at a glance.
  • Decide convergence with direct and limit comparison when no antiderivative exists.
  • Compute with the Gamma function and recognize Gaussian-type integrals.
  • Distinguish convergence, divergence, and the principal value.

What's Next

Chapter 18 turns integration loose on geometry and physics — areas between curves, volumes of revolution (Gabriel's Horn included), arc length, and work. Chapter 19 closes Part III with differential equations, where improper integrals and the Laplace transform return as solution tools. And the whole convergence toolkit reappears in Chapter 22 as the integral test.