$x$	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

16. $\int x e ^ { 2 x } d x =$
(A) $\frac { x e ^ { 2 x } } { 2 } - \frac { e ^ { 2 x } } { 4 } + C$
(B) $\frac { x e ^ { 2 x } } { 2 } - \frac { e ^ { 2 x } } { 2 } + C$
(C) $\frac { x e ^ { 2 x } } { 2 } + \frac { e ^ { 2 x } } { 4 } + C$
(D) $\frac { x e ^ { 2 x } } { 2 } + \frac { e ^ { 2 x } } { 2 } + C$
(E) $\frac { x ^ { 2 } e ^ { 2 x } } { 4 } + C$

43. $\int \arcsin x d x =$
(A) $\quad \sin x - \int \frac { x d x } { \sqrt { 1 - x ^ { 2 } } }$
(B) $\frac { ( \arcsin x ) ^ { 2 } } { 2 } + C$
(C) $\quad \arcsin x + \int \frac { d x } { \sqrt { 1 - x ^ { 2 } } }$
(D) $\quad x \arccos x - \int \frac { x d x } { \sqrt { 1 - x ^ { 2 } } }$
(E) $\quad x \arcsin x - \int \frac { x d x } { \sqrt { 1 - x ^ { 2 } } }$

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 $f(x)$.

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. $F(x) = \frac{2}{x} \mathrm{e}^{x^{2}}$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. We consider the function $f$ defined on $]0; +\infty[$ by: $f(x) = x\ln(x)$.
   Statement 1: $$\int_1^{\mathrm{e}} f(x)\,\mathrm{d}x = \frac{\mathrm{e}^2 + 1}{4}$$
   
2. Let $n$ and $k$ be two non-zero natural integers such that $k \leqslant n$.
   Statement 2: $$n \times \binom{n-1}{k-1} = k \times \binom{n}{k}$$
   
3. For the three following statements, we consider that space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
   Let $d$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= t + 1 \\ y &= 2t + 1 \\ z &= -t \end{array}\right., t \in \mathbb{R}$. Let $d'$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= 2t' - 1 \\ y &= -t' + 2 \\ z &= t' + 1 \end{array}\right., t' \in \mathbb{R}$. Let $P$ be the plane with Cartesian equation: $2x + y - 2z + 18 = 0$. Let A be the point with coordinates $(-1; -3; 2)$ and B be the point with coordinates $(-5; -5; 6)$. We call the perpendicular bisector plane of segment $[\mathrm{AB}]$ the plane passing through the midpoint of segment $[\mathrm{AB}]$ and perpendicular to the line $(\mathrm{AB})$.
   Statement 3: Point A belongs to line $d$. Statement 4: Lines $d$ and $d'$ are secant. Statement 5: Plane $P$ is the perpendicular bisector plane of segment $[\mathrm{AB}]$.

[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

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

What is the value of $\int _ { e } ^ { e ^ { 2 } } \frac { \ln x - 1 } { x ^ { 2 } } d x$? [3 points]
(1) $\frac { e - 2 } { e ^ { 2 } }$
(2) $\frac { e - 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { e + 1 } { e ^ { 2 } }$
(5) $\frac { e + 2 } { e ^ { 2 } }$

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.

The Euler Gamma function is defined, for all real $x > 0$, by: $$\Gamma(x) = \int_{0}^{+\infty} e^{-t} t^{x-1} dt$$ Calculate $\Gamma(n)$ for all natural integers $n$, $n \geqslant 1$.

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: $$\left| w_{k} - \frac{1}{12} \int_{k-1}^{k} \frac{\mathrm{~d}t}{t^{2}} \right| \leqslant \frac{1}{6} \int_{k-1}^{k} \frac{dt}{t^{3}}$$

For all integers $k \geqslant 2$, we denote: $$w_{k} = \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$ and $v_{n} = \sum_{k=n+1}^{+\infty} w_{k}$.
Deduce that $$\left| v_{n} - \frac{1}{12n} \right| \leqslant \frac{1}{12n^{2}}$$ then that: $$\ln n! = n \ln n - n + \frac{1}{2} \ln n + a + \frac{1}{12n} + O\left(\frac{1}{n^{2}}\right)$$

We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Show that, for all integers $n$, $n \geqslant 0$, $$\forall x > 0, \quad J_{n+1}(x) = \frac{n+1}{x} J_{n}(x+1)$$

We define for all real $x > 0$ the sequence $(J_{n}(x))_{n \geqslant 0}$ by: $$J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$ Deduce that, for all $x > 0$, $$J_{n}(x) = \frac{n!}{x(x+1) \cdots (x+n-1)(x+n)}$$

The function $h$ is defined on $\mathbb{R}$ by $$h(u) = u - [u] - 1/2$$ Using integration by parts, justify, for $x > 0$, the convergence of the following integral: $$\int_{0}^{+\infty} \frac{h(u)}{u+x} du$$

For every natural integer $k$ we set $$m_{k} = \frac{1}{2\pi} \int_{-2}^{2} x^{k} \sqrt{4 - x^{2}} \, \mathrm{d}x$$ Using integration by parts, show that, for every natural integer $k$, $$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$

For all $n \in \mathbb { N }$, set $\mu _ { n } = \left( X ^ { n } \mid 1 \right)$ with the inner product $$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Using integration by parts, show $$\forall n \in \mathbb { N } ^ { * } , \quad 4 \mu _ { n - 1 } - \mu _ { n } = \frac { 2 \times 4 ^ { n } } { \pi } \int _ { 0 } ^ { 1 } x ^ { n - 3 / 2 } ( 1 - x ) ^ { 3 / 2 } \mathrm { d } x = \frac { 3 } { 2 n - 1 } \mu _ { n } .$$

Prove that $$\int _ { 0 } ^ { + \infty } \frac { \sin ( x ) } { x } d x = \frac { \pi } { 2 }$$

Let $\left(a_{n}\right)_{n \geqslant 2}$ be a sequence of real numbers. For $t \in \mathbb{R}$, we set $A(t) = \sum_{2 \leqslant k \leqslant t} a_{k}$. Let $b : [2, +\infty[ \rightarrow \mathbb{R}$ be a function of class $\mathscr{C}^{1}$. Show that for any integer $n \geqslant 2$, $$\sum_{k=2}^{n} a_{k} b(k) = A(n) b(n) - \int_{2}^{n} b^{\prime}(t) A(t) \mathrm{d}t.$$

7. Integrate
(A) $\int \frac { x ^ { 3 } + 3 x + 2 } { \left( x ^ { 2 } + 1 \right) ^ { 2 } ( x + 1 ) } \mathrm { dx }$.
(B) $\quad \int _ { 0 } ^ { \pi } \frac { e ^ { \cos x } } { e ^ { \cos x } + e ^ { - \cos x } } \mathrm { dx }$. 8. Let $\mathrm { f } ( \mathrm { x } )$ be a continuous function given by
$$f ( x ) = \left\{ \begin{array} { c c } 2 x , & | x | \leq 1 \\ x ^ { 2 } + a x + b , & | x | > 1 \end{array} \right.$$
Find the area of the region in the third quadrant bounded by the curves $x = - 2 y 2$ and $y = \mathrm { f } ( \mathrm { x } )$ lying on the left on the line $8 \mathrm { x } + 1 = 0$. 9. Find the co-ordinates of all the P on the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { y } 2 / \mathrm { b } 2 = 1$, for which the area of the triangle PON is maximum, where O denotes the origin and N , the foot of the perpendicular from O to the tangent at P . 10. A curve passing through the point $( 1,1 )$ has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x -axis. Determine the equation of the curve. 11. Eight players $\mathrm { P } 1 , \mathrm { P } 2 , \ldots \ldots . \mathrm { P } 8$ play a knock-out tournament. It is known that whenever the players Pi and Pj play, the play Pi will win if $\mathrm { i } < \mathrm { j }$. Assuming that the players are paired at random in each round, what is the probability that the player P4 reaches the final? 12. Let $\vec { u }$ and $\vec { v }$ be unit vectors. If $\vec { w }$ is a vector such that $\vec { w } + ( \vec { w } \times \vec { u } ) = \vec { v }$, then prove that $| ( \vec { u } \times \vec { v } ) \cdot \vec { w } | \leq \frac { 1 } { 2 }$ and that the equality holds if and only if $\vec { u }$ is perpendicular to $\vec { v }$.