$\int _ { 1 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$ is
(A) $- \frac { 1 } { e }$
(B) $\frac { 1 } { 2 e }$
(C) $\frac { 1 } { e }$
(D) $\frac { 2 } { e }$
(E) divergent

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function: $$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$ is integrable on the interval $]-1,1[$.

Study the convergence of the improper integrals $$\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 - \cos \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 + \cos \theta ) d \theta$$
Deduce that, for all $x \in \mathbb { R }$, the integral $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ converges.

Show that the improper integral $\int _ { 0 } ^ { \pi / 2 } \ln ( \cos \theta ) d \theta$ converges.

We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number. b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$.
Show that $$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$

We call a distribution on $\mathcal{D}$ any linear map $T : \mathcal{D} \rightarrow \mathbb{R}$ which satisfies $$\forall \varphi \in \mathcal{D}, \forall (\varphi_n)_{n \in \mathbb{N}} \in \mathcal{D}^{\mathbb{N}} \quad \varphi_n \xrightarrow{\mathcal{D}} \varphi \Longrightarrow T(\varphi_n) \rightarrow T(\varphi)$$
Show that if $f \in \mathcal{F}_{sr}$ then the map $T_f$ defined by $$\forall \varphi \in \mathcal{D} \quad T_f(\varphi) = \int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x$$ defines a distribution on $\mathcal{D}$.

Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Justify that $U$ defines a distribution on $\mathcal{D}$.

Let $a$ be a real number. a) Show that the map $\delta_a$ which associates to every $\varphi \in \mathcal{D}$ the value $\varphi(a)$ is a distribution. b) Using the sequence of functions $(\varphi_n)_{n \in \mathbb{N}^*}$ of elements of $\mathcal{D}$ defined by $$\forall t \in \mathbb{R}, \varphi_n(t) = \begin{cases} \exp\left(\frac{(t-a)^2}{(t-a+1/n)(t-a-1/n)}\right) & \text{if } t \in ]a-1/n, a+1/n[ \\ 0 & \text{otherwise} \end{cases}$$ show that $\forall f \in \mathcal{F}_{sr}, T_f \neq \delta_a$.

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Show that the integral $I _ { p , q }$ exists for every pair of natural integers $( p , q )$.

Let $x > 0$. Show that $t \mapsto t ^ { x - 1 } e ^ { - t }$ is integrable on $] 0 , + \infty [$.

Throughout the rest of the problem, we denote $\Gamma$ the function defined on $\mathbb { R } ^ { + * }$ by $\Gamma ( x ) = \int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - t } \mathrm {~d} t$. We admit that $\Gamma$ is of class $\mathcal { C } ^ { \infty }$ on its domain of definition, takes strictly positive values and satisfies, for every real $x > 0$, the relation $\Gamma ( x + 1 ) = x \Gamma ( x )$.
Let $x$ and $\alpha$ be two strictly positive real numbers. Justify the existence of $\int _ { 0 } ^ { + \infty } t ^ { x - 1 } e ^ { - \alpha t } \mathrm {~d} t$ and give its value as a function of $\Gamma ( x )$ and $\alpha ^ { x }$.

For $( x , y )$ in $\left( \mathbb { R } ^ { + * } \right) ^ { 2 }$, we define $\beta ( x , y ) = \int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } \mathrm {~d} t$.
Justify the existence of $\beta ( x , y )$ for $x > 0$ and $y > 0$.

Let $f \in \mathcal{S}$. We assume that $\mathcal{F}(f)$ is integrable on $\mathbb{R}$.
Prove that $f(0) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) \mathrm{d}\xi$.
Deduce, using the function $h : t \mapsto f(x+t)$, that
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-\infty}^{+\infty} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$

Determine for which $x \in \mathbb{R}$ the integral below is convergent $$\int_{0}^{1} \frac{t^{x} - 1}{1 - t} \mathrm{~d}t$$

Let $(U_n)_{n \geqslant 1}$ be a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}$.
Justify the existence of $\int_0^1 \frac{t-1}{\ln t} \,\mathrm{d}t$ and then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n}) \,\mathrm{d}t$.

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $(U_n)_{n \geqslant 1}$ a sequence of mutually independent random variables following a Bernoulli distribution with parameter $1/2$. We set $$\forall n \in \mathbb{N}^{\star}, \quad Y_n = \sum_{k=1}^{n} \frac{U_k}{2^k}.$$ For every continuous function $f$ from $[0,1]$ to $\mathbb{R}$, the sequence $(\mathbb{E}(f(Y_n)))_{n \geqslant 1}$ converges to $\int_0^1 f(t)\,\mathrm{d}t$.
An application. Justify the existence of $\int_0^1 \frac{t-1}{\ln t}\,\mathrm{d}t$ then determine its value.
One may consider $\int_0^1 \mathbb{E}(t^{Y_n})\,\mathrm{d}t$.

Show that $$\lim_{a \rightarrow +\infty} \int_0^a |\sin(x^2)| \mathrm{d}x = +\infty$$

Show that $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \left| \sin \left( x ^ { 2 } \right) \right| \mathrm { d } x = + \infty$$

Let $\lambda > 0$ be fixed. We consider the space $\mathcal{C}(\mathbf{R}, \mathbf{R})$ of continuous functions from $\mathbf{R}$ to $\mathbf{R}$. We denote by $\mathcal{E}$ the vector subspace of $\mathcal{C}(\mathbf{R}, \mathbf{R})$ defined by $$\mathcal{E} = \left\{ f \in \mathcal{C}(\mathbf{R}, \mathbf{R}) \mid \exists (a, A) \in \left(\mathbf{R}_*^+\right)^2 \text{ such that } \forall y \in \mathbf{R},\ |f(y)| \leq A \exp\left(-y^2/a\right) \right\}$$
For all $(f, g) \in \mathcal{E}^2$, show that $fg$ is integrable on $\mathbf{R}$.

Show that the integral $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s$ is convergent.

We admit that $\int _ { 0 } ^ { + \infty } \operatorname { sinc } ( s ) \mathrm { d } s = \frac { \pi } { 2 }$. If $a$ and $b$ are two real numbers, we denote $K _ { a , b }$ the function defined for all real $t$ by $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Deduce the existence and value of $\lim _ { N \rightarrow + \infty } \int _ { - N } ^ { N } K _ { a , b } ( t ) \mathrm { d } t$ in the case where $a < b$.

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $\left. ]0,1 \right]$?

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. For which values of the real parameter $\alpha$ is the function $x \mapsto x ^ { \alpha } W ( x )$ integrable on $[ 1 , + \infty [$?

Let $P \in \mathbb { R } [ X ]$ and $Q \in \mathbb { R } [ X ]$. Show that the function $x \mapsto P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } }$ is integrable on $\left. ] 0,1 \right]$.

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $\int _ { \lfloor x \rfloor } ^ { x } \frac { q ( u ) } { u } \mathrm {~d} u$ tends to 0 as $x$ tends to $+ \infty$, and deduce the convergence of the integral $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u$, as well as the equality
$$\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { u } \mathrm {~d} u = \frac { \ln ( 2 \pi ) } { 2 } - 1$$