$\quad \int \sec x \tan x \, d x =$
(A) $\sec x + C$
(B) $\tan x + C$
(C) $\frac { \sec ^ { 2 } x } { 2 } + C$
(D) $\frac { \tan ^ { 2 } x } { 2 } + C$
(E) $\frac { \sec ^ { 2 } x \tan ^ { 2 } x } { 2 } + C$

We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 } { x } ( 1 + \ln x )$$

1. In the three situations below, we have drawn, in an orthonormal coordinate system, the representative curve $\mathscr { C } _ { f }$ of the function $f$ and a curve $\mathscr { C } _ { F }$. In only one situation, the curve $\mathscr { C } _ { F }$ is the representative curve of a primitive $F$ of the function $f$. Which one? Justify the answer.

We consider the functions $f(x) = x\mathrm{e}^{1-x^{2}}$ and $g(x) = \mathrm{e}^{1-x}$.

1. Find a primitive $F$ of the function $f$ on $\mathbb{R}$.
2. Deduce the value of $\int_{0}^{1} \left(\mathrm{e}^{1-x} - x\mathrm{e}^{1-x^{2}}\right) \mathrm{d}x$.
3. Give a graphical interpretation of this result.

Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by:
$$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { 1 - x } }$$

Part A

1. Study the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.
2. Prove that for all real $x$ in the interval $[ 0 ; 1 ] , f ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + \mathrm { e } }$ (recall that $\mathrm { e } = \mathrm { e } ^ { 1 }$ ).
3. Show then that $\int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x = \ln ( 2 ) + 1 - \ln ( 1 + \mathrm { e } )$.

Part B

Let $n$ be a natural number. Consider the functions $f _ { n }$ defined on $[ 0 ; 1 ]$ by:
$$f _ { n } ( x ) = \frac { 1 } { 1 + n \mathrm { e } ^ { 1 - x } }$$
We denote $\mathscr { C } _ { n }$ the representative curve of the function $f _ { n }$ in the plane with an orthonormal coordinate system. Consider the sequence with general term
$$u _ { n } = \int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$

1. The representative curves of the functions $f _ { n }$ for $n$ varying from 1 to 5 are drawn in the appendix. Complete the graph by drawing the curve $\mathscr { C } _ { 0 }$ representative of the function $f _ { 0 }$.
2. Let $n$ be a natural number, interpret graphically $u _ { n }$ and specify the value of $u _ { 0 }$.
3. What conjecture can be made regarding the direction of variation of the sequence $\left( u _ { n } \right)$ ?

Prove this conjecture.
4. Does the sequence ( $u _ { n }$ ) have a limit?

Exercise 1 -- Part B
The proportion of individuals who possess a certain type of equipment in a population is modelled by the function $p$ defined on $[ 0 ; + \infty [$ by $$p ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 0,2 x } } .$$ The real number $x$ represents the time elapsed, in years, since January 1st, 2000. The number $p ( x )$ models the proportion of equipped individuals after $x$ years. Thus, for this model, $p ( 0 )$ is the proportion of equipped individuals on January 1st, 2000 and $p ( 3.5 )$ is the proportion of equipped individuals in the middle of 2003.

1. What is, for this model, the proportion of equipped individuals on January 1st, 2010? Give a value rounded to the nearest hundredth.
2. 1. [a.] Determine the direction of variation of the function $p$ on $[ 0 ; + \infty [$.
   2. [b.] Calculate the limit of the function $p$ as $x \to + \infty$.
   3. [c.] Interpret this limit in the context of the exercise.
3. It is considered that, when the proportion of equipped individuals exceeds $95\%$, the market is saturated. Determine, by explaining the approach, the year in which this occurs.
4. The average proportion of equipped individuals between 2008 and 2010 is defined by $$m = \frac { 1 } { 2 } \int _ { 8 } ^ { 10 } p ( x ) \mathrm { d } x$$
   1. [a.] Verify that, for all real $x \geqslant 0$, $$p ( x ) = \frac { \mathrm { e } ^ { 0,2 x } } { 1 + \mathrm { e } ^ { 0,2 x } }$$
   2. [b.] Deduce an antiderivative of the function $p$ on $[ 0 ; + \infty [$.
   3. [c.] Determine the exact value of $m$ and its approximation to the nearest hundredth.

Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{x^2-3}$$ One of the antiderivatives $F$ of the function $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 2x\mathrm{e}^{x^2-3}$ b. $F(x) = \left(2x^2+1\right)\mathrm{e}^{x^2-3}$ c. $F(x) = \frac{1}{2}x\mathrm{e}^{x^2-3}$ d. $F(x) = \frac{1}{2}\mathrm{e}^{x^2-3}$

Define a function $f$ as follows: $f ( 0 ) = 0$ and, for any $x > 0$, $$f ( x ) = \lim _ { L \rightarrow \infty } \int _ { \frac { 1 } { x } } ^ { L } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$$ (or, in simpler notation, the improper integral $\int _ { \frac { 1 } { x } } ^ { \infty } \frac { 1 } { t ^ { 2 } } \cos ( t ) \, d t$).
(i) Show that the definition makes sense for any $x > 0$ by justifying why the limit in the definition exists, i.e., why the improper integral converges.
(ii) Find $f ^ { \prime } \left( \frac { 1 } { \pi } \right)$ if it exists. Clearly indicate the basic result(s) you are using.
(iii) Using the hint or otherwise, find $\lim _ { h \rightarrow 0 ^ { + } } \frac { f ( h ) - f ( 0 ) } { h }$, i.e., the right hand derivative of $f$ at $x = 0$. We can take the limit only from the right hand side because $f ( x )$ is undefined for negative values of $x$. Hint: Break $f ( h )$ into two terms by using a standard technique with an appropriate choice. Then separately analyze the resulting two terms in the derivative.

A continuous function $f ( x )$ satisfies
$$f ( x ) = e ^ { x ^ { 2 } } + \int _ { 0 } ^ { 1 } t f ( t ) d t$$
What is the value of $\int _ { 0 } ^ { 1 } x f ( x ) d x$? [3 points]
(1) $e - 2$
(2) $\frac { e - 1 } { 2 }$
(3) $\frac { e } { 2 }$
(4) $e - 1$
(5) $\frac { e + 1 } { 2 }$

For a function $f ( x )$ that is differentiable on $x > 0$, $$f ^ { \prime } ( x ) = 2 - \frac { 3 } { x ^ { 2 } } , \quad f ( 1 ) = 5$$ For a function $g ( x )$ that is differentiable on $x < 0$ and satisfies the following conditions, what is the value of $g ( - 3 )$? [4 points] (가) For all real numbers $x < 0$, $g ^ { \prime } ( x ) = f ^ { \prime } ( - x )$. (나) $f ( 2 ) + g ( - 2 ) = 9$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a function $f ( x )$, if $f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x$ and $f ( 0 ) = 3$, what is the value of $f ( 2 )$? [3 points]

What is the value of $\int_{0}^{10} \frac{x+2}{x+1}\, dx$? [3 points]
(1) $10 + \ln 5$
(2) $10 + \ln 7$
(3) $10 + 2\ln 3$
(4) $10 + \ln 11$
(5) $10 + \ln 13$

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$. The map $S$ is defined by $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
Show that $$\forall (P,Q) \in E^2, \quad S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$$

Show that, for any natural integer $p$, the function $x \mapsto x^{2p} \exp\left(-x^{2}\right)$ is integrable on $\mathbb{R}$.

From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define $$h(x) = \sqrt{|f(x) - f(x_0)|}$$ We admit that the bijection $h : [x_0, 1] \rightarrow [0, h(1)]$ admits an inverse application $h^{-1} : [0, h(1)] \rightarrow [x_0, 1]$ that is infinitely differentiable.
We admit the identities: $$\lim_{a \rightarrow +\infty} \int_0^a \sin(x^2) \mathrm{d}x = \lim_{a \rightarrow +\infty} \int_0^a \cos(x^2) \mathrm{d}x = \frac{\sqrt{2\pi}}{4}$$
Show that, as $t \rightarrow +\infty$, $$\int_{x_0}^1 \sin(tf(x)) \mathrm{d}x = \sin\left(tf(x_0) + \frac{\pi}{4}\right) \sqrt{\frac{\pi}{2tf''(x_0)}} + O\left(\frac{1}{t}\right)$$

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 $g : [ 0,1 ] \rightarrow \mathbb { R }$.
For all $x \in \left[ x _ { 0 } , 1 \right]$, we define $$h ( x ) = \sqrt { \left| f ( x ) - f \left( x _ { 0 } \right) \right| }$$ We admit that the bijection $$h : \left\{ \begin{array} { c c c } { \left[ x _ { 0 } , 1 \right] } & \rightarrow & { [ 0 , h ( 1 ) ] } \\ x & \mapsto & h ( x ) \end{array} \right.$$ admits an inverse map $h ^ { - 1 } : [ 0 , h ( 1 ) ] \rightarrow \left[ x _ { 0 } , 1 \right]$ that is infinitely differentiable.
Show that, as $t \rightarrow + \infty$, $$\int _ { x _ { 0 } } ^ { 1 } \sin ( t f ( x ) ) \mathrm { d } x = \sin \left( t f \left( x _ { 0 } \right) + \frac { \pi } { 4 } \right) \sqrt { \frac { \pi } { 2 t f ^ { \prime \prime } \left( x _ { 0 } \right) } } + O \left( \frac { 1 } { t } \right)$$

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
$$\int _ { 0 } ^ { 1 } \ln \left( \frac { 1 - e ^ { - t u } } { t } \right) \mathrm { d } u \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } - 1$$
You may begin by establishing that $x \mapsto \frac { 1 - e ^ { - x } } { x }$ is decreasing on $\mathbf { R } _ { + } ^ { * }$.

Show that $$\int_{0}^{1} \ln\left(\frac{1-e^{-tu}}{t}\right) \mathrm{d}u \underset{t \rightarrow 0^+}{\longrightarrow} -1.$$ One may begin by establishing that $x \mapsto \frac{1-e^{-x}}{x}$ is decreasing on $\mathbf{R}_+$.

Show that the integral
$$\int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { 2 p + 1 } } { t ^ { 2 } } \mathrm {~d} t$$
converges and that:
$$\int _ { 0 } ^ { + \infty } \frac { 1 - ( \cos ( t ) ) ^ { 2 p + 1 } } { t ^ { 2 } } \mathrm {~d} t = ( 2 p + 1 ) \int _ { 0 } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t$$

We define $E _ { 1 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p = q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
Show that, for all $( p , q ) \in E _ { 1 }$, $$S _ { p , q } = \frac { \ln 2 } { p }$$

We define $E _ { 2 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p < q , p \mid q \right\}$ and $S _ { p , q } = \int _ { 0 } ^ { 1 } \dfrac { t ^ { q - 1 } } { 1 + t ^ { p } } d t$.
For all pairs $( p , q ) \in E _ { 2 }$, show that there exists a constant $\lambda := \lambda ( p , q )$ which one will determine, such that $$S _ { p , q } = \frac { ( - 1 ) ^ { \lambda - 1 } } { p } \left( \ln ( 2 ) - \sum _ { k = 1 } ^ { \lambda - 1 } \frac { ( - 1 ) ^ { k - 1 } } { k } \right)$$

For each positive integer $n$, define a function $f_n$ on $[0,1]$ as follows: $$f _ { n } ( x ) = \left\{ \begin{array} { c c c } 0 & \text { if } & x = 0 \\ \sin \frac { \pi } { 2 n } & \text { if } & 0 < x \leq \frac { 1 } { n } \\ \sin \frac { 2 \pi } { 2 n } & \text { if } & \frac { 1 } { n } < x \leq \frac { 2 } { n } \\ \sin \frac { 3 \pi } { 2 n } & \text { if } & \frac { 2 } { n } < x \leq \frac { 3 } { n } \\ \vdots & \vdots & \vdots \\ \sin \frac { n \pi } { 2 n } & \text { if } & \frac { n - 1 } { n } < x \leq 1 . \end{array} \right.$$ Then, the value of $\lim _ { n \rightarrow \infty } \int _ { 0 } ^ { 1 } f _ { n } ( x ) d x$ is
(A) $\pi$
(B) 1
(C) $\frac{1}{\pi}$
(D) $\frac{2}{\pi}$

Let $f : ( 0 , \infty ) \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in ( 0 , \infty )$, $$f ( 2 x ) = f ( x )$$ Show that the function $g$ defined by the equation $$g ( x ) = \int _ { x } ^ { 2 x } f ( t ) \frac { d t } { t } \text { for } x > 0$$ is a constant function.

Let $$f ( x ) = e ^ { - | x | } , x \in \mathbb { R }$$ and $$g ( \theta ) = \int _ { - 1 } ^ { 1 } f \left( \frac { x } { \theta } \right) d x , \theta \neq 0$$ Then, $$\lim _ { \theta \rightarrow 0 } \frac { g ( \theta ) } { \theta }$$ (A) equals 0 .
(B) equals $+ \infty$.
(C) equals 2 .
(D) does not exist.

Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\int_{-1}^{1} \psi(x)\,\mathrm{d}x = 1$. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Then $$\lim_{\varepsilon \rightarrow 0} \frac{1}{\varepsilon} \int_{1-\varepsilon}^{1+\varepsilon} f(y)\,\psi\!\left(\frac{1-y}{\varepsilon}\right) \mathrm{d}y$$ equals
(A) $f(1)$
(B) $f(1)\psi(0)$
(C) $f'(1)\psi(0)$
(D) $f(1)\psi(1)$