Let $g$ be the piecewise-linear function defined on $[-2\pi, 4\pi]$ whose graph is given above, and let $f(x) = g(x) - \cos\left(\dfrac{x}{2}\right)$.
(a) Find $\int_{-2\pi}^{4\pi} f(x)\, dx$. Show the computations that lead to your answer.
(b) Find all $x$-values in the open interval $(-2\pi, 4\pi)$ for which $f$ has a critical point.
(c) Let $h(x) = \int_{0}^{3x} g(t)\, dt$. Find $h^{\prime}\!\left(-\dfrac{\pi}{3}\right)$.

The graph of $f$ is shown above for $0 \leq x \leq 4$. What is the value of $\int _ { 0 } ^ { 4 } f ( x ) d x$ ?
(A) - 1
(B) 0
(C) 2
(D) 6
(E) 12

$\int _ { 1 } ^ { 4 } t ^ { - 3 / 2 } d t =$
(A) - 1
(B) $- \frac { 7 } { 8 }$
(C) $- \frac { 1 } { 2 }$
(D) $\frac { 1 } { 2 }$
(E) 1

If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	- 4
5	- 3
7	0

(B)

$x$	$f ( x )$
4	- 4
5	- 2
7	5

(C)

$x$	$f ( x )$
4	- 4
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

4. We define the number $I = \int _ { 0 } ^ { 1 } f _ { 1 } ( x ) \mathrm { d } x$.
Show that $I = \ln \left( \frac { 1 + \mathrm { e } } { 2 } \right)$. Give a graphical interpretation of $I$.

Part B

In this part, we choose $k = - 1$ and we wish to sketch the curve $\mathscr { C } _ { - 1 }$ representing the function $f _ { - 1 }$. For all real $x$, we call $P$ the point on $\mathscr { C } _ { 1 }$ with abscissa $x$ and $M$ the point on $\mathscr { C } _ { - 1 }$ with abscissa $x$. We denote by $K$ the midpoint of segment [ $M P$ ].

1. Show that, for all real $x , f _ { 1 } ( x ) + f _ { - 1 } ( x ) = 1$.
2. Deduce that point $K$ belongs to the line with equation $y = \frac { 1 } { 2 }$.
3. Sketch the curve $\mathscr { C } _ { - 1 }$ on the APPENDIX, to be returned with your answer sheet.
4. Deduce the area, in square units, of the region bounded by the curves $\mathscr { C } _ { 1 } , \mathscr { C } _ { - 1 }$, the $y$-axis and the line with equation $x = 1$.

Part C

In this part, we do not privilege any particular value of the parameter $k$. For each of the following statements, say whether it is true or false and justify your answer.

1. Whatever the value of the real number $k$, the graph of the function $f _ { k }$ is strictly between the lines with equations $y = 0$ and $y = 1$.
2. Whatever the value of the real $k$, the function $f _ { k }$ is strictly increasing.
3. For all real $u _ { n }$ & 4502 & 13378 & 39878 & 119122 & 356342 & 1066978 & 3196838 & 9582322 & 28730582 \hline \end{tabular}
   b. What conjecture can be made concerning the monotonicity of the sequence $\left( u _ { n } \right)$ ?

The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part B: study of Proposal B
This proposal is characterized by the following two conditions:

• the line with endpoints A and E is a portion of the graph of the function $f$ defined for all real $x \geqslant 0$ by: $f(x) = \ln(2x + 1)$;
• the line with endpoints B and G is a portion of the graph of the function $g$ defined for all real $x > 0$ by: $g(x) = k\left(\frac{1 - x}{x}\right)$, where $k$ is a positive real number to be determined.

1. a) Determine the abscissa of point E. b) Determine the value of the real number $k$, knowing that the abscissa of point G is equal to 0.5.
   
2. a) Prove that the function $f$ has as a primitive the function $F$ defined for all real $x \geqslant 0$ by: $$F(x) = (x + 0.5) \times \ln(2x + 1) - x.$$ b) Prove that $r = \frac{\mathrm{e}}{2} - 1$.
   
3. Determine a primitive $G$ of the function $g$ on the interval $]0; +\infty[$.
   
4. It is admitted that the previous results allow us to establish that $s = [\ln(2)]^2 + \frac{\ln(2) - 1}{2}$. Does Proposal B satisfy the conditions imposed by the manufacturer?

A homeowner wants to have a water tank built. This water tank must comply with the following specifications:

• it must be located two metres from his house;
• the maximum depth must be two metres;
• it must measure five metres long;
• it must follow the natural slope of the land.

The curved part is modelled by the curve $\mathscr{C}_f$ of the function $f$ on the interval $[2; 2e]$ defined by: $$f(x) = x \ln\left(\frac{x}{2}\right) - x + 2$$
The curve $\mathscr{C}_f$ is represented in an orthonormal coordinate system with unit $1\mathrm{m}$ and constitutes a profile view of the tank. We consider the points $\mathrm{A}(2; 2)$, $\mathrm{I}(2; 0)$ and $\mathrm{B}(2\mathrm{e}; 2)$.
Part A
The objective of this part is to evaluate the volume of the tank.

1. Justify that the points B and I belong to the curve $\mathscr{C}_f$ and that the x-axis is tangent to the curve $\mathscr{C}_f$ at point I.
2. We denote by $\mathscr{T}$ the tangent to the curve $\mathscr{C}_f$ at point B, and D the point of intersection of the line $\mathscr{T}$ with the x-axis. a. Determine an equation of the line $\mathscr{T}$ and deduce the coordinates of D. b. We call $S$ the area of the region bounded by the curve $\mathscr{C}_f$, the lines with equations $y = 2$, $x = 2$ and $x = 2\mathrm{e}$. $S$ can be bounded by the area of triangle ABI and that of trapezoid AIDB. What bounds on the volume of the tank can we deduce?
3. a. Show that, on the interval $[2; 2\mathrm{e}]$, the function $G$ defined by $$G(x) = \frac{x^2}{2} \ln\left(\frac{x}{2}\right) - \frac{x^2}{4}$$ is an antiderivative of the function $g$ defined by $g(x) = x \ln\left(\frac{x}{2}\right)$. b. Deduce an antiderivative $F$ of the function $f$ on the interval $[2; 2\mathrm{e}]$. c. Determine the exact value of the area $S$ and deduce an approximate value of the volume $V$ of the tank to the nearest $\mathrm{m}^3$.

Part B
For any real number $x$ between 2 and $2\mathrm{e}$, we denote by $v(x)$ the volume of water, expressed in $\mathrm{m}^3$, in the tank when the water level in the tank is equal to $f(x)$. We admit that, for any real number $x$ in the interval $[2; 2\mathrm{e}]$, $$v(x) = 5\left[\frac{x^2}{2}\ln\left(\frac{x}{2}\right) - 2x\ln\left(\frac{x}{2}\right) - \frac{x^2}{4} + 2x - 3\right]$$

1. What volume of water, to the nearest $\mathrm{m}^3$, is in the tank when the water level in the tank is one metre?
2. We recall that $V$ is the total volume of the tank, $f$ is the function defined at the beginning of the exercise and $v$ the function defined in Part B. We consider the following algorithm:

   \begin{tabular}{l} Variables:

   Processing:

   &

   $a$ is a real number

   $b$ is a real number

   $a$ takes the value 2

   $b$ takes the value $2\mathrm{e}$

   While $v(b) - v(a) > 10^{-3}$ do:

   $c$ takes the value $(a + b)/2$

   If $v(c) < V/2$, then:

   $a$ takes the value $c$

   Otherwise

   $b$ takes the value $c$

   End If

   End While

   Display $f(c)$

   \hline \end{tabular} Interpret the result that this algorithm allows to display.

Let $a$ be a real number between 0 and 1. We denote by $f _ { a }$ the function defined on $\mathbf { R }$ by:
$$f _ { a } ( x ) = a \mathrm { e } ^ { a x } + a .$$
We denote by $I ( a )$ the integral of the function $f _ { a }$ between 0 and 1:
$$I ( a ) = \int _ { 0 } ^ { 1 } f _ { a } ( x ) \mathrm { d } x$$

1. In this question, we set $a = 0$. Determine $I ( 0 )$.
2. In this question, we set $a = 1$. We therefore study the function $f _ { 1 }$ defined on $\mathbf { R }$ by: $$f _ { 1 } ( x ) = \mathrm { e } ^ { x } + 1$$ a. Without detailed study, sketch the graph of the function $f _ { 1 }$ on your paper in an orthogonal coordinate system and show the number $I ( 1 )$. b. Calculate the exact value of $I ( 1 )$, then round to the nearest tenth.
3. Does there exist a value of $a$ for which $I ( a )$ equals 2? If so, give an interval of width $10 ^ { - 2 }$ containing this value.

The purpose of the problem is the study of the integrals $I$ and $J$ defined by:
$$I = \int_{0}^{1} \frac{1}{1 + x} \mathrm{~d}x \quad \text{and} \quad J = \int_{0}^{1} \frac{1}{1 + x^{2}} \mathrm{~d}x$$
Part A: exact value of the integral $I$

1. Give a geometric interpretation of the integral $I$.
2. Calculate the exact value of $I$.

Part B: estimation of the value of $J$
Let $g$ be the function defined on the interval $[0; 1]$ by $g(x) = \frac{1}{1 + x^{2}}$. We denote $\mathscr{C}_{g}$ its representative curve in an orthonormal frame of the plane. We therefore have: $J = \int_{0}^{1} g(x) \mathrm{d}x$. The purpose of this part is to evaluate the integral $J$ using the probabilistic method described below. We choose at random a point $\mathrm{M}(x; y)$ by drawing independently its coordinates $x$ and $y$ at random according to the uniform distribution on $[0; 1]$. We admit that the probability $p$ that a point drawn in this manner is located below the curve $\mathscr{C}_{g}$ is equal to the integral $J$. In practice, we initialize a counter $c$ to 0, we fix a natural number $n$ and we repeat $n$ times the following process:

• we choose at random and independently two numbers $x$ and $y$, according to the uniform distribution on $[0; 1]$;
• if $\mathrm{M}(x; y)$ is below the curve $\mathscr{C}_{g}$ we increment the counter $c$ by 1.

We admit that $f = \frac{c}{n}$ is an approximate value of $J$. This is the principle of the so-called Monte-Carlo method.

The two parts of this exercise are independent.

Part A

Let the function $f$ defined on the set of real numbers by
$$f ( x ) = 2 \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C }$ its representative curve in an orthonormal coordinate system. We admit that, for all $x$ belonging to $[ 0 ; \ln ( 2 ) ] , f ( x )$ is positive. Indicate whether the following proposition is true or false by justifying your answer.
Proposition A: The area of the region bounded by the lines with equations $x = 0$ and $x = \ln ( 2 )$, the $x$-axis and the curve $\mathscr { C }$ is equal to 1 unit of area.

Part B

Let $n$ be a strictly positive integer. Let the function $f _ { n }$ defined on the set of real numbers by
$$f _ { n } ( x ) = 2 n \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C } _ { n }$ its representative curve in an orthonormal coordinate system. We admit that $f _ { n }$ is differentiable and that $\mathscr { C } _ { n }$ admits a horizontal tangent at a unique point $S _ { n }$. Indicate whether the following proposition is true or false by justifying your answer.
Proposition B: For all strictly positive integer $n$, the ordinate of the point $S _ { n }$ is $n ^ { 2 }$.

Consider the function $f$ defined on the interval $[0; +\infty[$ by $f(x) = k\mathrm{e}^{-kx}$ where $k$ is a strictly positive real number. We call $\mathcal{C}_f$ its graph in the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider point A on the curve $\mathcal{C}_f$ with x-coordinate 0 and point B on the curve $\mathcal{C}_f$ with x-coordinate 1. Point C has coordinates $(1; 0)$.

1. Determine an antiderivative of function $f$ on the interval $[0; +\infty[$.
2. Express, as a function of $k$, the area of triangle OCB and that of the region $\mathcal{D}$ bounded by the y-axis, the curve $\mathcal{C}_f$ and the segment $[OB]$.
3. Show that there exists a unique value of the strictly positive real number $k$ such that the area of region $\mathcal{D}$ is twice that of triangle OCB.

Part C

In this part, we consider the function $h$ defined on $\mathbb { R }$ by $$h ( x ) = ( x - 1 ) \mathrm { e } ^ { - 2 x } + 1 .$$ We admit that the function $h$ is differentiable on $\mathbb { R }$. We place ourselves in an orthonormal coordinate system ( O ; I, J). We denote $\mathscr { C } _ { h }$ the representative curve of the function $h$ and $d$ the line with equation $y = x$. We admit that the curve $\mathscr { C } _ { h }$ is above the line $d$ on the interval $[ 0 ; 1 ]$. Let $\mathscr { D }$ be the region of the plane bounded by the curve $\mathscr { C } _ { h }$, the line $d$ and the vertical lines with equations $x = 0$ and $x = 1$. Let $\mathscr { A }$ be the area of $\mathscr { D }$ expressed in square units.

1. On ANNEX 1, shade the region $\mathscr { D }$ and justify that $$\mathscr { A } = \int _ { 0 } ^ { 1 } [ h ( x ) - x ] \mathrm { d } x$$
2. a. Prove that, for all real $x$, $$h ( x ) - x = ( 1 - x ) \left( 1 - \mathrm { e } ^ { - 2 x } \right) .$$ b. We admit that, for all real $x$, $\mathrm { e } ^ { - 2 x } \geqslant 1 - 2 x$. Prove that, for all real $x$ in the interval $[ 0 ; 1 ]$, $$h ( x ) - x \leqslant 2 x - 2 x ^ { 2 } .$$ c. Deduce that $\mathscr { A } \leqslant \frac { 1 } { 3 }$.
3. Let $H$ be the function defined on $[ 0 ; 1 ]$ by $$H ( x ) = \frac { 1 } { 4 } ( 1 - 2 x ) \mathrm { e } ^ { - 2 x } + x$$ We admit that the function $H$ is an antiderivative of the function $h$ on $[ 0 ; 1 ]$. Determine the exact value of $\mathscr { A }$.

Let $a$ be a strictly positive real number. We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = a\ln(x)$$ We denote $\mathscr{C}$ its representative curve in an orthonormal coordinate system. Let $x_0$ be a real number strictly greater than 1.

1. Determine the abscissa of the point of intersection of the curve $\mathscr{C}$ and the x-axis.
2. Verify that the function $F$ defined by $F(x) = a[x\ln(x) - x]$ is a primitive of the function $f$ on the interval $]0; +\infty[$.
3. Deduce the area of the blue region as a function of $a$ and $x_0$.

We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point $M$ with abscissa $x_0$. We call $A$ the point of intersection of the tangent line $T$ with the y-axis and $B$ the orthogonal projection of $M$ onto the y-axis.

1. Prove that the length AB is equal to a constant (that is, to a number that does not depend on $x_0$) which we will determine. The candidate will take care to make their approach explicit.

A integral $\int_0^2 (3x^2 + 2x)\,dx$ é igual a
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16

QUESTION 141
A cylindrical tank has a base radius of 2 m and a height of 5 m. The volume of this tank, in cubic meters, is
(A) $10\pi$
(B) $15\pi$
(C) $20\pi$
(D) $25\pi$
(E) $30\pi$

QUESTION 159
The value of $\int_0^2 (3x^2 + 2x)\, dx$ is
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

The value of $\displaystyle\int_0^2 (3x^2 + 2x)\,dx$ is:
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square.
(a) Show how to write a definite integral that will calculate the volume of this cave.
(b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?

Write the values of the following.
(a) $\int_{-3}^{3} \left| 3x^{2} - 3 \right| dx$.
(b) $f'(1)$ where $f(t) = \int_{0}^{t} \left| 3x^{2} - 3 \right| dx$.

Statements
(13) $\lim _ { x \rightarrow 0 } e ^ { \frac { 1 } { x } } = + \infty$. (14) The following inequality is true. $$\lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { 100 } } < \lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { \frac { 1 } { 100 } } }$$ (15) For any positive integer $n$, $$\int _ { - n } ^ { n } x ^ { 2023 } \cos ( n x ) \, dx < \frac { n } { 2023 }$$ (16) There is no polynomial $p ( x )$ for which there is a single line that is tangent to the graph of $p ( x )$ at exactly 100 points.

The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$.
$$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$
Find the value of $a ^ { 4 }$. [3 points]

(Calculus) Find the length of the curve $y = \frac { 1 } { 3 } \left( x ^ { 2 } + 2 \right) ^ { \frac { 3 } { 2 } }$ from $x = 0$ to $x = 6$. [4 points]

When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points]
(1) $- 4$
(2) $- 2$
(3) 0
(4) 2
(5) 4

A quadratic function $f ( x )$ satisfies $f ( 0 ) = - 1$ and
$$\int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x$$
What is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 10
(3) 9
(4) 8
(5) 7

Find the value of $\int _ { 0 } ^ { 5 } ( 4 x - 3 ) d x$. [3 points]