$x$	2	5
$f ( x )$	4	7
$f ^ { \prime } ( x )$	2	3

The table above gives values of the differentiable function $f$ and its derivative $f ^ { \prime }$ at selected values of $x$. If $\int _ { 2 } ^ { 5 } f ( x ) \, d x = 14$, what is the value of $\int _ { 2 } ^ { 5 } x \cdot f ^ { \prime } ( x ) \, d x$?
(A) 13
(B) 27
(C) $\frac { 63 } { 2 }$
(D) 41

Let $f$ be the function defined for $x > 0$, with $f(e) = 2$ and $f'$, the first derivative of $f$, given by $f'(x) = x^2 \ln x$.
(a) Write an equation for the line tangent to the graph of $f$ at the point $(e, 2)$.
(b) Is the graph of $f$ concave up or concave down on the interval $1 < x < 3$? Give a reason for your answer.
(c) Use antidifferentiation to find $f(x)$.

The function $f$ is twice differentiable for $x > 0$ with $f ( 1 ) = 15$ and $f ^ { \prime \prime } ( 1 ) = 20$. Values of $f ^ { \prime }$, the derivative of $f$, are given for selected values of $x$ in the table above.

$x$	1	1.1	1.2	1.3	1.4
$f ^ { \prime } ( x )$	8	10	12	13	14.5

(a) Write an equation for the line tangent to the graph of $f$ at $x = 1$. Use this line to approximate $f ( 1.4 )$.
(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to approximate $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$. Use the approximation for $\int _ { 1 } ^ { 1.4 } f ^ { \prime } ( x ) d x$ to estimate the value of $f ( 1.4 )$. Show the computations that lead to your answer.
(c) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 1.4 )$.

Let $f$ be a differentiable function such that $\int f ( x ) \sin x \, d x = - f ( x ) \cos x + \int 4 x ^ { 3 } \cos x \, d x$. Which of the following could be $f ( x )$ ?
(A) $\cos x$
(B) $\sin x$
(C) $4 x ^ { 3 }$
(D) $- x ^ { 4 }$
(E) $x ^ { 4 }$

Exercise 1 -- Common to all candidates

Part A

We consider the function $h$ defined on the interval $[ 0 ; + \infty [$ by: $$h ( x ) = x \mathrm { e } ^ { - x }$$

1. Determine the limit of the function $h$ at $+ \infty$.
2. Study the variations of the function $h$ on the interval $[ 0 ; + \infty [$ and draw up its table of variations.
3. The objective of this question is to determine a primitive of the function $h$. a. Verify that for every real number $x$ belonging to the interval $[ 0 ; + \infty [$, we have: $$h ( x ) = \mathrm { e } ^ { - x } - h ^ { \prime } ( x )$$ where $h ^ { \prime }$ denotes the derivative function of $h$. b. Determine a primitive on the interval $[ 0 ; + \infty [$ of the function $x \longmapsto \mathrm { e } ^ { - x }$. c. Deduce from the two previous questions a primitive of the function $h$ on the interval $[ 0 ; + \infty [$.

Part B

We define the functions $f$ and $g$ on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x } + \ln ( x + 1 ) \quad \text { and } \quad g ( x ) = \ln ( x + 1 )$$ We denote $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ the respective graphical representations of the functions $f$ and $g$ in an orthonormal coordinate system.

1. For a real number $x$ belonging to the interval $[ 0 ; + \infty [$, we call $M$ the point with coordinates $( x ; f ( x ) )$ and $N$ the point with coordinates $( x ; g ( x ) )$: $M$ and $N$ are therefore the points with abscissa $x$ belonging respectively to the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$. a. Determine the value of $x$ for which the distance $MN$ is maximum and give this maximum distance. b. Place on the graph provided in the appendix the points $M$ and $N$ corresponding to the maximum value of $MN$.
2. Let $\lambda$ be a real number belonging to the interval $[ 0 ; + \infty [$. We denote $D _ { \lambda }$ the region of the plane bounded by the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ and by the lines with equations $x = 0$ and $x = \lambda$. a. Shade the region $D _ { \lambda }$ corresponding to the value $\lambda$ proposed on the graph in the appendix. b. We denote $A _ { \lambda }$ the area of the region $D _ { \lambda }$, expressed in square units. Prove that: $$A _ { \lambda } = 1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } .$$ c. Calculate the limit of $A _ { \lambda }$ as $\lambda$ tends to $+ \infty$ and interpret the result.
3. We consider the following algorithm: \begin{verbatim} Variables: $\lambda$ is a positive real number $S$ is a real number strictly between 0 and 1. Initialization: Input $S$ $\lambda$ takes the value 0 Processing: While $1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } < S$ do $\lambda$ takes the value $\lambda + 1$ End While Output: Display $\lambda$ \end{verbatim} a. What value does this algorithm display if we input the value $S = 0.8$? b. What is the role of this algorithm?

The flow of water from a tap has a constant and moderate flow rate.
We are particularly interested in a part of the flow profile represented in appendix 1 by the curve $C$ in an orthonormal coordinate system.

Part A

We consider that the curve $C$ given in appendix 1 is the graphical representation of a function $f$ differentiable on the interval $] 0 ; 1 ]$ which respects the following three conditions:
$$( H ) : f ( 1 ) = 0 \quad f ^ { \prime } ( 1 ) = 0.25 \quad \text { and } \lim _ { \substack { x \rightarrow 0 \\ x > 0 } } f ( x ) = - \infty .$$

1. Can the function $f$ be a polynomial function of degree two? Why?
2. Let $g$ be the function defined on the interval $]0 ; 1]$ by $g ( x ) = k \ln x$. a. Determine the real number $k$ so that the function $g$ respects the three conditions $( H )$. b. Does the representative curve of the function $g$ coincide with the curve $C$ ? Why?
3. Let $h$ be the function defined on the interval $]0; 1]$ by $h ( x ) = \frac { a } { x ^ { 4 } } + b x$ where $a$ and $b$ are real numbers. Determine $a$ and $b$ so that the function $h$ respects the three conditions ( $H$ ).

Part B

We admit in this part that the curve $C$ is the graphical representation of a function $f$ continuous, strictly increasing, defined and differentiable on the interval $] 0 ; 1 ]$ with expression:
$$f ( x ) = \frac { 1 } { 20 } \left( x - \frac { 1 } { x ^ { 4 } } \right)$$

1. Justify that the equation $f ( x ) = - 5$ admits on the interval $] 0 ; 1 ]$ a unique solution which will be denoted $\alpha$. Determine an approximate value of $\alpha$ to $10 ^ { - 2 }$ near.
2. It is admitted that the volume of water in $\mathrm { cm } ^ { 3 }$, contained in the first 5 centimetres of the flow, is given by the formula: $V = \int _ { \alpha } ^ { 1 } \pi x ^ { 2 } f ^ { \prime } ( x ) \mathrm { d } x$. a. Let $u$ be the function differentiable on $] 0; 1]$ defined by $u ( x ) = \frac { 1 } { 2 x ^ { 2 } }$. Determine its derivative function. b. Determine the exact value of $V$. Using the approximate value of $\alpha$ obtained in question 1, give an approximate value of $V$.

Let $f$ be the function defined on $] 0 ; + \infty \left[ \text{ by } f ( x ) = x ^ { 2 } \ln x \right.$.
A primitive $F$ of $f$ on $] 0$; $+ \infty [$ is defined by: a. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } \left( \ln x - \frac { 1 } { 3 } \right)$; b. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } ( \ln x - 1 )$; c. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 }$; d. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 } ( \ln x - 1 )$.

Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x \mathrm{e}^{x}$.
A primitive $F$ on $\mathbb{R}$ of the function $f$ is defined by:
A. $F(x) = \frac{x^{2}}{2} \mathrm{e}^{x}$
B. $F(x) = (x - 1) \mathrm{e}^{x}$
C. $F(x) = (x + 1) \mathrm{e}^{x}$
D. $F(x) = \frac{2}{x} \mathrm{e}^{x^{2}}$.

In this part, we consider that the function $f$, defined and twice differentiable on $[0; +\infty[$, is defined by
$$f(x) = (4x - 2)\mathrm{e}^{-x + 1}.$$
We will denote respectively $f'$ and $f''$ the derivative and second derivative of the function $f$.

1. Study of the function $f$ a. Show that $f'(x) = (-4x + 6)\mathrm{e}^{-x + 1}$. b. Use this result to determine the complete table of variations of the function $f$ on $[0; +\infty[$. It is admitted that $\lim_{x \rightarrow +\infty} f(x) = 0$. c. Study the convexity of the function $f$ and specify the abscissa of any possible inflection point of the representative curve of $f$.
2. We consider a function $F$ defined on $[0; +\infty[$ by $F(x) = (ax + b)\mathrm{e}^{-x + 1}$, where $a$ and $b$ are two real numbers. a. Determine the values of the real numbers $a$ and $b$ such that the function $F$ is a primitive of the function $f$ on $[0; +\infty[$. b. It is admitted that $F(x) = (-4x - 2)\mathrm{e}^{-x + 1}$ is a primitive of the function $f$ on $[0; +\infty[$. Deduce the exact value, then an approximate value to $10^{-2}$ near, of the integral $$I = \int_{\frac{3}{2}}^{8} f(x)\mathrm{d}x.$$
3. A municipality has decided to build a freestyle scooter track. The profile of this track is given by the representative curve of the function $f$ on the interval $[\frac{3}{2}; 8]$. The unit of length is the meter. a. Give an approximate value to the nearest cm of the height of the starting point D. b. The municipality has organized a graffiti competition to decorate the wall profile of the track. The selected artist plans to cover approximately $75\%$ of the wall surface. Knowing that a 150 mL aerosol can covers a surface of $0.8\mathrm{~m}^2$, determine the number of cans she will need to use to create this work.

Exercise 2 (5 points)

Part A

A craftsman creates chocolate candies whose shape recalls the profile of the local mountain. The base of such a candy is modeled by the shaded surface defined in an orthonormal coordinate system with unit 1 cm.
This surface is bounded by the x-axis and the graph denoted $\mathscr { C } _ { f }$ of the function $f$ defined on $[ - 1 ; 1 ]$ by: $$f ( x ) = \left( 1 - x ^ { 2 } \right) \mathrm { e } ^ { x } .$$
The objective of this part is to calculate the volume of chocolate needed to manufacture a chocolate candy.

1. a. Justify that for all $x$ belonging to the interval $[ - 1 ; 1 ]$ we have $f ( x ) \geqslant 0$. b. Show using integration by parts that: $$\int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x = 2 \int _ { - 1 } ^ { 1 } x \mathrm { e } ^ { x } \mathrm {~d} x .$$
2. The volume $V$ of chocolate, in $\mathrm { cm } ^ { 3 }$, needed to manufacture a candy is given by: $$V = 3 \times S$$ where $S$ is the area, in $\mathrm { cm } ^ { 2 }$, of the colored surface. Deduce that this volume, rounded to $0.1 \mathrm {~cm} ^ { 3 }$, equals $4.4 \mathrm {~cm} ^ { 3 }$.

Part B

We now consider the profit realized by the craftsman on the sale of these chocolate candies as a function of the weekly sales volume.
This profit can be modeled by the function $B$ defined on the interval $[ 0.01 ; + \infty [$ by: $$B ( q ) = 8 q ^ { 2 } [ 2 - 3 \ln ( q ) ] - 3 .$$
The profit is expressed in tens of euros and the quantity $q$ in hundreds of candies. We admit that the function $B$ is differentiable on $\left[ 0.01 ; + \infty \left[ \right. \right.$. We denote $B ^ { \prime }$ its derivative function.

1. a. Determine $\lim _ { q \rightarrow + \infty } B ( q )$. b. Show that, for all $q \geqslant 0.01 , B ^ { \prime } ( q ) = 8 q ( 1 - 6 \ln ( q ) )$. c. Study the sign of $B ^ { \prime } ( q )$, and deduce the direction of variation of $B$ on $[ 0.01 ; + \infty [$. Draw the complete variation table of function $B$. d. What is the maximum profit, to the nearest euro, that the craftsman can expect?
2. a. Show that the equation $B ( q ) = 10$ has a unique solution $\beta$ on the interval $[1.2; + \infty [$. Give an approximate value of $\beta$ to $10 ^ { - 3 }$ near. b. We admit that the equation $B ( q ) = 10$ has a unique solution $\alpha$ on $[ 0.01 ; 1.2 [$. We are given $\alpha \approx 0.757$. Deduce the minimum and maximum number of chocolate candies to sell to achieve a profit greater than 100 euros.

A certification body is commissioned to evaluate two heating devices, one from brand A and the other from brand B.
Parts 1 and 2 are independent.
Part 1: device from brand A Using a probe, the temperature inside the combustion chamber of a brand A device was measured. Below is a representation of the temperature curve in degrees Celsius inside the combustion chamber as a function of time elapsed, expressed in minutes, since the combustion chamber was ignited.
By reading the graph:

1. Give the time at which the maximum temperature is reached inside the combustion chamber.
2. Give an approximate value, in minutes, of the duration during which the temperature inside the combustion chamber exceeds $300 ^ { \circ } \mathrm { C }$.
3. We denote by $f$ the function represented on the graph. Estimate the value of $\frac { 1 } { 600 } \int _ { 0 } ^ { 600 } f ( t ) \mathrm { d } t$. Interpret the result.

Part 2: study of a function Let the function $g$ be defined on the interval $[0 ; + \infty [$ by: $$g ( t ) = 10 t \mathrm { e } ^ { - 0.01 t } + 20 .$$

1. Determine the limit of $g$ at $+ \infty$.
2. a. Show that for all $t \in \left[ 0 ; + \infty \left[ , \quad g ^ { \prime } ( t ) = ( - 0.1 t + 10 ) \mathrm { e } ^ { - 0.01 t } \right. \right.$. b. Study the variations of the function $g$ on $[0 ; + \infty [$ and construct its variation table.
3. Prove that the equation $g ( t ) = 300$ has exactly two distinct solutions on $[0 ; + \infty [$. Give approximate values to the nearest integer.
4. Using integration by parts, calculate $\int _ { 0 } ^ { 600 } g ( t ) \mathrm { d } t$.

Part 3: evaluation For a brand B device, the temperature in degrees Celsius inside the combustion chamber $t$ minutes after ignition is modelled on $[0 ; 600]$ by the function $g$.
The certification body awards one star for each criterion validated among the following four:

• Criterion 1: the maximum temperature is greater than $320 ^ { \circ } \mathrm { C }$.
• Criterion 2: the maximum temperature is reached in less than 2 hours.
• Criterion 3: the average temperature during the first 10 hours after ignition exceeds $250 ^ { \circ } \mathrm { C }$.
• Criterion 4: the temperature inside the combustion chamber must not exceed $300 ^ { \circ } \mathrm { C }$ for more than 5 hours.

Does each device obtain exactly three stars? Justify your answer.

We consider $n$ a non-zero natural integer. We consider the function $f_n$ defined on the interval $[0; 1]$ by:
$$f_n(x) = x^n e^{1-x}$$
We admit that the function $f_n$ is differentiable on $[0; 1]$ and we denote $f_n'$ its derivative function.

Part A

In this part we study the case where $n = 1$. We thus study the function $f_1$ defined on $[0; 1]$ by:
$$f_1(x) = x e^{1-x}$$

1. Show that $f_1'(x)$ is strictly positive for all real $x$ in $[0; 1[$.
2. Deduce the table of variations of the function $f_1$ on the interval $[0; 1]$.
3. Deduce that the equation $f_1(x) = 0.1$ admits a unique solution in the interval $[0; 1]$.

Part B

We consider the sequence $(u_n)$ defined for all non-zero natural integers $n$ by
$$u_n = \int_0^1 f_n(x) \, dx \quad \text{that is} \quad u_n = \int_0^1 x^n e^{1-x} \, dx$$
We admit that $u_1 = e - 2$.

1. a. Justify that for all $x \in [0; 1]$ and for all non-zero natural integers $n$, $$0 \leq x^{n+1} \leq x^n$$ b. Deduce that for all non-zero natural integers $n$, $$0 \leq u_{n+1} \leq u_n.$$ c. Show that the sequence $(u_n)$ is convergent.
2. a. Using integration by parts, prove that for all non-zero natural integers $n$ we have: $$u_{n+1} = (n+1)u_n - 1$$ b. Consider the Python script below defining the function suite(): \begin{verbatim} from math import exp def suite(): u = ... for n in range (1, ...): u = ... return \end{verbatim} Copy and complete the Python script above so that the function suite() returns the value of $\int_0^1 x^8 e^{1-x} \, dx$.
3. a. Prove that for all non-zero natural integers $n$ we have: $$u_n \leq \frac{e}{n+1}$$ b. Deduce the limit of the sequence $(u_n)$.

We equip the plane with an orthonormal coordinate system. For every natural integer $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by: $$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geqslant 1,\ f_n(x) = x^n \mathrm{e}^{-x}.$$ For every natural integer $n$, we denote $\mathscr{C}_n$ the representative curve of the function $f_n$.
Parts A and B are independent.
Part A: Study of the functions $f_n$ for $n \geqslant 1$
We consider a natural integer $n \geqslant 1$.

1. a. We admit that the function $f_n$ is differentiable on $[0; +\infty[$. Show that for all $x \geqslant 0$, $$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}.$$ b. Justify all elements of the table below:
   

   $x$	0		$n$		$+\infty$
   $f_n'(x)$		+	0	-
   			$\left(\frac{n}{\mathrm{e}}\right)^n$
   $f_n$
   	0				0

   
   
2. Justify by calculation that the point $\mathrm{A}\left(1; \mathrm{e}^{-1}\right)$ belongs to the curve $\mathscr{C}_n$.

Part B: Study of the integrals $\int_0^1 f_n(x)\,\mathrm{d}x$ for $n \geqslant 0$
In this part, we study the functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural integer $n$ by: $$I_n = \int_0^1 f_n(x)\,\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\,\mathrm{d}x.$$

1. On the graph in APPENDIX, the curves $\mathscr{C}_0, \mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_{10}$ and $\mathscr{C}_{100}$ are represented. a. Give a graphical interpretation of $I_n$. b. By reading this graph, what conjecture can be made about the limit of the sequence $(I_n)$?
2. Calculate $I_0$.
3. a. Let $n$ be a natural integer. Prove that for all $x \in [0; 1]$, $$0 \leqslant x^{n+1} \leqslant x^n.$$ b. Deduce that for every natural integer $n$, we have: $$0 \leqslant I_{n+1} \leqslant I_n.$$
4. Prove that the sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we will denote $\ell$.
5. Using integration by parts, prove that for every natural integer $n$ we have: $$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}.$$
6. a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction. b. Prove that $\ell = 0$. You may use question 6.a.
7. The script of the \texttt{mystere} function is given below, written in Python language. The constant \texttt{e} has been imported. \begin{verbatim} def mystere(n): I = 1 - 1/e L = [I] for i in range(n): I = (i + 1)*I - 1/e L.append(I) return L \end{verbatim} What does \texttt{mystere(100)} return in the context of the exercise?

We denote by $f$ the function defined on the interval $[ 0 ; \pi ]$ by
$$f ( x ) = \mathrm { e } ^ { x } \sin ( x )$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in a coordinate system.

PART A

1. a. Prove that for every real number $x$ in the interval $[ 0 ; \pi ]$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { x } [ \sin ( x ) + \cos ( x ) ]$$ b. Justify that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$
2. a. Determine an equation of the tangent $T$ to the curve $\mathscr { C } _ { f }$ at the point with abscissa 0. b. Prove that the function $f$ is convex on the interval $\left[ 0 ; \frac { \pi } { 2 } \right]$. c. Deduce that for every real number $x$ in the interval $\left[ 0 ; \frac { \pi } { 2 } \right] , \mathrm { e } ^ { x } \sin ( x ) \geqslant x$.
3. Justify that the point with abscissa $\frac { \pi } { 2 }$ of the representative curve of the function $f$ is an inflection point.

PART B

We denote
$$I = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \sin ( x ) \mathrm { d } x \text { and } \quad J = \int _ { 0 } ^ { \frac { \pi } { 2 } } \mathrm { e } ^ { x } \cos ( x ) \mathrm { d } x$$

1. By integrating by parts the integral $I$ in two different ways, establish the following two relations: $$I = 1 + J \quad \text { and } \quad I = \mathrm { e } ^ { \frac { \pi } { 2 } } - J$$
2. Deduce that $I = \frac { 1 + \mathrm { e } ^ { \frac { \pi } { 2 } } } { 2 }$.
3. We denote by $g$ the function defined on $\mathbb { R }$ by $g ( x ) = x$. Calculate the exact value of the area of the shaded region situated between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$ and the lines with equations $x = 0$ and $x = \frac { \pi } { 2 }$.

Calculate the following two definite integrals. It may be useful to first sketch the graph. $$\int_{1}^{e^{2}} \ln|x|\, dx \qquad \int_{-1}^{1} \frac{\ln|x|}{|x|}\, dx$$

[Calculus] For two functions $f ( x )$ and $g ( x )$ that have second derivatives on the set of all real numbers, consider the definite integral $$\int _ { 0 } ^ { 1 } \left\{ f ^ { \prime } ( x ) g ( 1 - x ) - g ^ { \prime } ( x ) f ( 1 - x ) \right\} d x$$ Let the value of this integral be $k$. Which of the following statements in are correct? [4 points]
ㄱ. $\int _ { 0 } ^ { 1 } \left\{ f ( x ) g ^ { \prime } ( 1 - x ) - g ( x ) f ^ { \prime } ( 1 - x ) \right\} d x = - k$ ㄴ. If $f ( 0 ) = f ( 1 )$ and $g ( 0 ) = g ( 1 )$, then $k = 0$. ㄷ. If $f ( x ) = \ln \left( 1 + x ^ { 4 } \right)$ and $g ( x ) = \sin \pi x$, then $k = 0$.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a positive number $a$, the function $f ( x ) = \int _ { 0 } ^ { x } ( a - t ) e ^ { t } \, d t$ has a maximum value of 32. Find the area enclosed by the curve $y = 3 e ^ { x }$ and the two lines $x = a$ and $y = 3$. [4 points]

Two polynomial functions $f ( x ) , g ( x )$ satisfy for all real numbers $x$ $$f ( - x ) = - f ( x ) , \quad g ( - x ) = g ( x )$$ For the function $h ( x ) = f ( x ) g ( x )$, $$\int _ { - 3 } ^ { 3 } ( x + 5 ) h ^ { \prime } ( x ) d x = 10$$ What is the value of $h ( 3 )$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies $$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$ When the function $F ( x )$ is defined as $$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$ what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points]
(1) $4 - \sqrt { 2 }$
(2) $2 + \sqrt { 2 }$
(3) $5 - \sqrt { 2 }$
(4) $1 + 2 \sqrt { 2 }$
(5) $2 + 2 \sqrt { 2 }$

For a real number $t$, define the function $f ( x )$ as $$f ( x ) = \left\{ \begin{array} { c c } 1 - | x - t | & ( | x - t | \leq 1 ) \\ 0 & ( | x - t | > 1 ) \end{array} \right.$$ For a certain odd number $k$, the function $$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$ satisfies the following condition.
When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$. Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]

Find the value of $\int _ { 0 } ^ { \pi } x \cos ( \pi - x ) d x$. [3 points]

A function $f ( x )$ that is increasing and differentiable on the set of all real numbers satisfies the following conditions. (가) $f ( 1 ) = 1 , \int _ { 1 } ^ { 2 } f ( x ) d x = \frac { 5 } { 4 }$ (나) When the inverse function of $f ( x )$ is $g ( x )$, for all real numbers $x \geq 1$, $g ( 2 x ) = 2 f ( x )$. When $\int _ { 1 } ^ { 8 } x f ^ { \prime } ( x ) d x = \frac { q } { p }$, find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.) [4 points]

We propose to calculate the area $\mathscr{A}$ of the domain $\mathscr{H}$ of $\mathbb{R}^2$ containing all the points $w(n, t)$ when $n$ ranges over $\mathbb{N}^*$ and $t$ ranges over $I = \left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$. This domain is bounded by two parametrized arcs defined by $$z : I \rightarrow \mathbb{C}, t \mapsto \rho(t) \mathrm{e}^{\mathrm{i}\theta(t)} = \sqrt{1 + 3\sin^2 t}\, \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$ $$v : I \rightarrow \mathbb{C}, t \mapsto \sqrt{1 + 3\sin^2 t}\left(1 + \frac{1}{4}\sin\left(\frac{2}{3}\left(t - \frac{\pi}{4}\right)\right)\right) \mathrm{e}^{\mathrm{i}\left(\arctan(2\tan t) + \pi \mathrm{E}\left(\frac{t}{\pi} + \frac{1}{2}\right)\right)}$$
III.D.1) Recall the statement of the Green-Riemann theorem. Explain how this theorem translates in the case of an area calculation. III.D.2) Recall the formula giving the scalar product of two complex numbers. Deduce the expression of the scalar product $\langle u \circ v(t), v'(t) \rangle$, when $u$ and $v$ are the applications $u : \mathbb{C} \rightarrow \mathbb{C}, z \mapsto \mathrm{i}z$ and $v : t \mapsto \sigma(t) \mathrm{e}^{\mathrm{i}\mu(t)}$, where $\sigma$ and $\mu$ are two functions defined on an interval $J$ of $\mathbb{R}$, with real values and of class $C^1$. III.D.3) If $d(t) = \arctan(2\tan(t))$, simplify $\frac{1}{2}\left(1 + 3\sin^2 t\right) d'(t)$. III.D.4) Deduce from the previous questions an expression of $\mathscr{A}$ in the form of an integral. Simplify this integral using the identity obtained in III.D.3). Finally, calculate $\mathscr{A}$.

For all integers $k \geqslant 2$, we set: $$u_{k} = \ln k - \int_{k-1}^{k} \ln t \, dt$$ Using two integrations by parts, show that: $$u_{k} = \frac{1}{2}(\ln k - \ln(k-1)) - \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ Using yet another integration by parts, show that: $$\left| w_{k} - \frac{1}{12} \int_{k-1}^{k} \frac{\mathrm{~d}t}{t^{2}} \right| \leqslant \frac{1}{6} \int_{k-1}^{k} \frac{dt}{t^{3}}$$