We fix $n \in \mathbb { N }$. We define the linear map: $$\begin{aligned} \Delta : \mathbb { R } [ X ] & \rightarrow \mathbb { R } [ X ] \\ P ( X ) & \mapsto P ( X + 1 ) - P ( X ) \end{aligned}$$
Using an enclosure by integrals, determine an asymptotic equivalent of $U _ { n } ( p ) = \sum _ { k = 0 } ^ { p } k ^ { n }$, with $n \geqslant 1$ fixed, as $p$ tends to $+ \infty$.

Show that $g_{\sigma}$ is integrable on $\mathbb{R}$.

Assuming that $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \mathrm{d}x = \sqrt{\pi}$, give the value of $\int_{-\infty}^{+\infty} g_{\sigma}(x) \mathrm{d}x$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.

Let $x \in \mathcal{D}_{\zeta}$ and let $n \in \mathbb{N}$ such that $n \geqslant 2$. Show: $$\int_{n}^{n+1} \frac{\mathrm{~d}t}{t^{x}} \leqslant \frac{1}{n^{x}} \leqslant \int_{n-1}^{n} \frac{\mathrm{~d}t}{t^{x}}$$

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.

Using the result of Q5, deduce that for all $x \in \mathcal{D}_{\zeta}$, $$1 + \frac{1}{(x-1)2^{x-1}} \leqslant \zeta(x) \leqslant 1 + \frac{1}{x-1}$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Show $$\forall n \in \llbracket 2, +\infty\llbracket, \forall x \in \mathbb{R}, \quad \left(1 - \frac{4x^2}{n^2}\right) I_n(x) = \frac{n-1}{n} I_{n-2}(x) \quad \text{and} \quad \left(1 - \frac{4x^2}{n^2}\right) \frac{I_n(x)}{I_n(0)} = \frac{I_{n-2}(x)}{I_{n-2}(0)}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. By differentiating $x \mapsto \ln(\cos(\pi x))$, show $$\forall x \in J, \quad \pi \tan(\pi x) = -\frac{2I_{4n}^{\prime}(2x)}{I_{4n}(2x)} + \frac{I_{2n}^{\prime}(x)}{I_{2n}(x)} + \sum_{k=1}^{n} \frac{8x}{(2k-1)^2} \frac{1}{1 - \frac{4x^2}{(2k-1)^2}}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the inequality $t\cos(t) \leqslant \sin(t)$ for $t \in [0, \pi/2]$, deduce $$\forall n \in \mathbb{N}^{\star}, \forall x \in [0,1], \quad 0 \leqslant -I_n^{\prime}(x) \leqslant \frac{4x}{n} I_n(x)$$ then, for $x \in [0,1]$, the limit $\lim_{n \rightarrow +\infty} \frac{I_n^{\prime}(x)}{I_n(x)}$.

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 $[ 1 , + \infty [$?

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

Show that the limits $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x \quad \text { and } \lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \cos \left( x ^ { 2 } \right) \mathrm { d } x$$ exist and are finite.

From now on, $f$ denotes an infinitely differentiable function from $[ 0,1 ]$ to $\mathbb { R }$. We assume that there exists a unique point $x _ { 0 } \in \left[ 0,1 \left[ \right. \right.$ where $f ^ { \prime }$ vanishes. We also assume that $f ^ { \prime \prime } \left( x _ { 0 } \right) > 0$. We are also given an infinitely differentiable function $g : [ 0,1 ] \rightarrow \mathbb { R }$.
Show that we have, as $t \rightarrow + \infty$, $$\int _ { x _ { 0 } } ^ { 1 } g ( x ) \sin ( t f ( x ) ) \mathrm { d } x = g \left( x _ { 0 } \right) \int _ { x _ { 0 } } ^ { 1 } \sin ( t f ( x ) ) \mathrm { d } x + O \left( \frac { 1 } { t } \right)$$

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 $a < b$.

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 $K _ { a , b } ( t ) = \begin{cases} \frac { \mathrm { e } ^ { \mathrm { i } t b } - \mathrm { e } ^ { \mathrm { i } t a } } { 2 \mathrm { i } t } & \text { if } t \neq 0 , \\ \frac { b - a } { 2 } & \text { if } t = 0 . \end{cases}$ Let $X : \Omega \rightarrow \mathbb { R }$ be a random variable such that $X ( \Omega )$ is finite. We assume that the real numbers $a$ and $b$ do not belong to $X ( \Omega )$. Show that $$\frac { 1 } { \pi } \int _ { - N } ^ { N } \phi _ { X } ( - t ) K _ { a , b } ( t ) \mathrm { d } t \xrightarrow [ N \rightarrow + \infty ] { } \mathbb { P } ( a < X < b )$$

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ For $k \in \mathbb{N}$, what is the value of $m_{2k+1}$?

Let $( m , n ) \in \mathbb { N } ^ { 2 }$. Calculate $\int _ { 0 } ^ { \pi / 2 } \sin ( ( 2 m + 1 ) \theta ) \sin ( ( 2 n + 1 ) \theta ) \mathrm { d } \theta$.

Show that, for all $\alpha \in \mathbb { R } _ { + } ^ { * } , p _ { \alpha }$ belongs to $E$, 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, and $p_\alpha$ is the function $t \mapsto t^\alpha$.

Let $P$ be a polynomial function not identically zero with real coefficients. Show that the restriction of $P$ to $\mathbb { R } _ { + } ^ { * }$ belongs to $E$ if and only if $P ( 0 ) = 0$, 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.

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 $\int _ { 1 } ^ { + \infty } \frac { q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u$ is well-defined for all real $t > 0$.

Show that $\int_{1}^{+\infty} \frac{q(u)}{e^{tu}-1} \mathrm{~d}u$ is well defined for all real $t > 0$.