Proof of The Integral of |x| exp(−ax²/2) over ℝ

Routine facts about real numbers are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, Toolkit for CA1c, Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (ii)), and so is the agreement of the earlier and current versions of the Borel $\sigma$-algebra, the integral of nonnegative functions and integrable functions (The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (v), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (vi)). Riemann integrals $\int_{u}^{v}g$ of restrictions $g|_{[u,v]}$, and $I_{n}(g)=\int_{-n}^{n}g$ for $n\in\mathbb{N}$, are as in Single-Variable Calculus on an Interval.

An antiderivative, and the continuity of $h_{a}$. Let $E:\mathbb{R}\to\mathbb{R}$ be the function $E(x)=\exp(-ax^{2}/2)$, the composite of $x\mapsto-ax^{2}/2$ with $\exp$ (Continuity of Identity Maps, Compositions and Reciprocals (ii)), and let $F:\mathbb{R}\to\mathbb{R}$ be the function $F(x)=-E(x)/a$. The map $x\mapsto-ax^{2}/2$ is differentiable at every point with derivative $-ax$, by the rules for the identity, products and scalar multiples (Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (vi), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v)). The derivative of $\exp$ (Basic Properties of the Exponential Function (vii)), the chain rule (Chain Rule for Real Functions on Intervals (i)) and the rule for scalar multiples then give, for every $x\in\mathbb{R}$,

$$E'(x)=-ax\exp(-ax^{2}/2),\qquad F'(x)=x\exp(-ax^{2}/2).$$

Being differentiable at every point, $E$ and $F$ are continuous on $\mathbb{R}$ (The Increment Form of the Derivative, and Differentiability Implies Continuity (ii)). Now $h_{a}$ is the product of $E$ with $x\mapsto|x|$, which is continuous by The Absolute Value on the Real Line is 1-Lipschitz and Continuous (ii), so $h_{a}$ is continuous on $\mathbb{R}$ (Continuity of Sums and Products of Real-Valued Functions on a Metric Space (v)); and $h_{a}$ is nonnegative.

The Riemann integrals. Let $n\in\mathbb{N}$ be arbitrary. The restrictions of $h_{a}$ to $[0,n]$ and to $[-n,0]$ are continuous (Restriction Stability of Continuity and of the Derivative (i)), hence Riemann integrable (A Continuous Function on a Closed Interval is Riemann Integrable (i)). The restrictions of the antiderivative $F$ to these intervals serve on both of them: they are continuous, and differentiable with the derivative of $F$ (Restriction Stability of Continuity and of the Derivative (i), Restriction Stability of Continuity and of the Derivative (ii)); hence $-F|_{[-n,0]}$ is continuous and differentiable with derivative $-(F|_{[-n,0]})'$, by the rules for scalar multiples (Continuity of Sums and Products of Real-Valued Functions on a Metric Space (iv), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v)). Since $|x|=x$ for $x\ge0$ and $|x|=-x$ for $x\le0$, this gives

$$(F|_{[0,n]})'(x)=x\exp(-ax^{2}/2)=h_{a}(x)\quad(0\le x\le n),\qquad(-F|_{[-n,0]})'(x)=-x\exp(-ax^{2}/2)=h_{a}(x)\quad(-n\le x\le0).$$

The fundamental theorem of calculus (Fundamental Theorem of Calculus, Part II: Evaluating an Integral by an Antiderivative (i)), with $F(0)=-1/a$ as $\exp(0)=1$ (Basic Properties of the Exponential Function (i)), therefore gives

$$\int_{0}^{n}h_{a}=F(n)-F(0)=\frac{1-\exp(-an^{2}/2)}{a},\qquad\int_{-n}^{0}h_{a}=-F(0)+F(-n)=\frac{1-\exp(-an^{2}/2)}{a}.$$

The restriction of $h_{a}$ to $[-n,n]$ is continuous as well (Restriction Stability of Continuity and of the Derivative (i)), hence Riemann integrable, in particular bounded (A Continuous Function on a Closed Interval is Riemann Integrable (i)), so additivity over subintervals (Additivity of the Riemann Integral over Adjacent Intervals, Subintervals and Oriented Integrals, and Integrals of Restrictions (i)) gives

$$I_{n}(h_{a})=\frac{2}{a}\bigl(1-\exp(-an^{2}/2)\bigr).$$

The supremum. The function $h_{a}$ lies in $\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, as The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (a result resting on dependent choice) states for every continuous nonnegative function, and by The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (ii)

$$\int_{\mathbb{R}}h_{a}\,d\lambda=\sup_{n\in\mathbb{N}}I_{n}(h_{a}).$$

Since $\exp$ is positive (Basic Properties of the Exponential Function (iii)), the formula of The Riemann integrals shows $I_{n}(h_{a})\le2/a$ for every $n$. For the reverse bound, let $n\in\mathbb{N}$. The lower bound $\exp(u)\ge1+u$ (Basic Properties of the Exponential Function (v)) gives $\exp(an^{2}/2)\ge1+an^{2}/2>an^{2}/2$, and taking reciprocals (Basic Properties of the Exponential Function (iv)) gives $\exp(-an^{2}/2)<2/(an^{2})$, so

$$\frac{2}{a}-I_{n}(h_{a})=\frac{2}{a}\exp(-an^{2}/2)<\frac{4}{a^{2}n^{2}}.$$

Now let $\varepsilon>0$ be given. Only then choose, by The Integers in the Real Numbers: Greatest Elements, the Integer Part, and Archimedean Consequences (vii) applied to the real number $1+4/(a^{2}\varepsilon)$, a natural number $N$ with

$$N>1+\frac{4}{a^{2}\varepsilon}.$$

In particular $N>1$, so $N^{2}\ge N$, and $N>4/(a^{2}\varepsilon)$; hence $4/(a^{2}N^{2})\le4/(a^{2}N)<\varepsilon$ and $I_{N}(h_{a})>2/a-\varepsilon$. Thus $2/a$ is an upper bound of the $I_{n}(h_{a})$ that no smaller number is, so the supremum is $2/a$ (Order and Suprema in the Extended Real Line (iv)). Together with $h_{a}\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, this proves (i).

Integrability. Since $\sup_{n}I_{n}(h_{a})=2/a<\infty$, the integrability criterion for nonnegative continuous functions (The Lebesgue Integral over ℝ of a Nonnegative Continuous Function as the Limit of Its Riemann Integrals over [−n, n] (iii)) gives $h_{a}\in\mathcal{L}^{1}(\lambda)$, which is (ii).