Show that $\int_{\lfloor x \rfloor}^{x} \frac{q(u)}{u} \mathrm{~d}u$ tends to 0 when $x$ 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 $$\int_{1}^{+\infty} \frac{q(u)}{u} \mathrm{~d}u = \frac{\ln(2\pi)}{2} - 1$$

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$. Show that $h _ { \alpha }$ is a continuous decreasing integrable function on $[ 0,1 [$.

We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ Using the result of Question 7, deduce that $f \in E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

Justify the existence of $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$ and give the exact value of $K _ { 1 }$.

Study the variations of the function $t \mapsto t \ln(t)$ on $\mathbf{R}_{+}^{*}$. Verify that the function can be extended by continuity at 0 and verify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x) \varphi(x) \mathrm{d}x = 1$.
Recall that for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$, the entropy of $f$ with respect to $\varphi$ is defined by: $$\operatorname{Ent}_{\varphi}(f) = \int_{-\infty}^{+\infty} \ln(f(x)) f(x) \varphi(x) \mathrm{d}x$$

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show, using the dominated convergence theorem, that:
$$\lim _ { \theta \rightarrow \pi ^ { - } } g ( \theta ) \sin ( x \theta ) = \int _ { - \infty } ^ { + \infty } \frac { \mathrm { d } u } { 1 + u ^ { 2 } }$$

Show that the function $\varphi : t \longmapsto \frac{1}{\sqrt{t}}$ is integrable on $]0,1[$, then show that the function $\varphi$ belongs to $\mathscr{D}_{0,1}$.

Consider the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$. Deduce from the previous questions that the function $h$ belongs to $\mathscr{D}_{0,1}$.

Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$.
Let $y \in ] 0,1 [$. Justify that the function
$$x \mapsto \frac { x ^ { r } y ^ { s } } { 1 - x y }$$
is integrable on $[ 0,1 ]$.

Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. For $y \in ] 0,1 [$, we set
$$f _ { r , s } ( y ) = \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x$$
Show that $f _ { r , s }$ is continuous and integrable on $] 0,1 [$.

Let $r$ and $s$ be two strictly positive natural integers such that $r \geqslant s$. We set
$$J _ { r , s } = \int _ { 0 } ^ { 1 } f _ { r , s } ( y ) \mathrm { d } y = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { r } y ^ { s } } { 1 - x y } \mathrm {~d} x \mathrm {~d} y$$
Show that
$$J _ { r , s } = \sum _ { k = 0 } ^ { + \infty } \frac { 1 } { ( r + k + 1 ) ( s + k + 1 ) }$$

Show that $\int_{-\infty}^{+\infty} \mathrm{e}^{-\frac{x^2}{2}} \mathrm{~d}x$ converges. We admit that its value is $\sqrt{2\pi}$.

Let $a$ and $\beta$ be two positive real numbers. For every integer $n > 0$, define $a_n = \int_{\beta}^{n} \frac{a}{u(u^{a}+2+u^{-a})} du$. Then $\lim_{n \to \infty} a_n$ is equal to
(a) $1/(1+\beta^{a})$
(b) $\beta^{a}/(1+\beta^{-a})$
(c) $\beta^{a}/(1+\beta^{a})$
(d) $\beta^{-a}/(1+\beta^{a})$

For $n \in \mathbb { N }$, let $a _ { n }$ be defined as $$a _ { n } = \int _ { 0 } ^ { n } \frac { 1 } { 1 + n x ^ { 2 } } d x$$ Then $\lim _ { n \rightarrow \infty } a _ { n }$
(A) equals 0
(B) equals $\frac { \pi } { 4 }$
(C) equals $\frac { \pi } { 2 }$
(D) does not exist