For $n \in \mathbb{N}$, we set $I_{n} = \int_{0}^{+\infty} x^{n} e^{-x}\, dx$.
Show that for $n \geqslant 1$, we have $$I_{n} = \left(\frac{n}{e}\right)^{n} \sqrt{n} \int_{-\sqrt{n}}^{+\infty} \left(1 + \frac{x}{\sqrt{n}}\right)^{n} e^{-x\sqrt{n}}\, dx.$$

We admit the identity $$\int _ { - \infty } ^ { + \infty } \exp \left( - x ^ { 2 } \right) \mathrm { d } x = \sqrt { \pi }$$
(a) Show that for all integer $n \in \mathbb { N }$, we have $$n ! = \int _ { 0 } ^ { + \infty } e ^ { - t } t ^ { n } \mathrm {~d} t$$
(b) Using the preceding results, recover Stirling's formula giving an asymptotic equivalent of $n !$.

Show that for all $a \in \mathbb { R }$, $$\int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \sum _ { n = 0 } ^ { + \infty } ( - 1 ) ^ { n } \frac { a ^ { 4 n + 3 } } { ( 2 n + 1 ) ! ( 4 n + 3 ) }$$

We admit the identities: $$\lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \lim _ { a \rightarrow + \infty } \int _ { 0 } ^ { a } \cos \left( x ^ { 2 } \right) \mathrm { d } x = \frac { \sqrt { 2 \pi } } { 4 }$$
Show that there exist real numbers $c , c ^ { \prime } \in \mathbb { R }$ such that, as $a \rightarrow + \infty$, we have $$\int _ { 0 } ^ { a } \sin \left( x ^ { 2 } \right) \mathrm { d } x = \frac { \sqrt { 2 \pi } } { 4 } + \frac { c } { a } \cos \left( a ^ { 2 } \right) + \frac { c ^ { \prime } } { a ^ { 3 } } \sin \left( a ^ { 2 } \right) + O \left( \frac { 1 } { a ^ { 5 } } \right) .$$

Using a series expansion under the integral, show that
$$\int _ { 0 } ^ { + \infty } \ln \left( 1 - e ^ { - u } \right) \mathrm { d } u = - \frac { \pi ^ { 2 } } { 6 }$$

Using a series expansion under the integral, show that $$\int_{0}^{+\infty} \ln(1-e^{-u}) \mathrm{d}u = -\frac{\pi^2}{6}$$

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 $\mathbf { R } _ { + } ^ { * }$.

Prove, without using what precedes, that
$$\int _ { 0 } ^ { + \infty } \frac { x e ^ { - x } } { 1 - e ^ { - x } } \mathrm {~d} x = \frac { \pi ^ { 2 } } { 6 }$$

Show that, for all $k \in \mathbb { N } , \int _ { 0 } ^ { + \infty } t ^ { k } \mathrm { e } ^ { - t } \mathrm {~d} t = k !$.

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$. Show that $h _ { \alpha }$ is a continuous decreasing integrable function on $[ 0,1 [$.

For any real $\alpha > 0$, consider the function $h _ { \alpha } : t \mapsto \ln \left( \frac { 1 - t ^ { 2 } } { \alpha ^ { 2 } + t ^ { 2 } } \right)$ and set $J _ { \alpha } = \int _ { 0 } ^ { 1 } h _ { \alpha } ( t ) \, \mathrm { d } t$. Justify that $$J _ { \alpha } = \int _ { 0 } ^ { 1 } \ln ( 1 - t ) \, \mathrm { d } t + \int _ { 0 } ^ { 1 } \ln ( 1 + t ) \, \mathrm { d } t - \int _ { 0 } ^ { 1 } \ln \left( \alpha ^ { 2 } + t ^ { 2 } \right) \mathrm { d } t = \int _ { 0 } ^ { 2 } \ln ( u ) \, \mathrm { d } u - \int _ { 0 } ^ { 1 } \ln \left( \alpha ^ { 2 } + t ^ { 2 } \right) \mathrm { d } t.$$

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.

5. Deduce the value of $\int _ { 0 } ^ { \infty } \frac { \sin t } { t } \mathrm {~d} t$.

Problem 2: linear recurrent sequences

We denote by $\mathrm { M } _ { d } ( \mathbb { C } )$ the space of $d \times d$ square matrices with complex coefficients and we identify $\mathbb { C } ^ { d }$ with the space of column vectors of size $d$.
For a vector $x = \left( x _ { 1 } , \ldots , x _ { d } \right) \in \mathbb { C } ^ { d }$, we define $\| x \| _ { \infty } = \max _ { 1 \leqslant i \leqslant d } \left| x _ { i } \right|$ and $\| x \| _ { 1 } = \sum _ { i = 1 } ^ { d } \left| x _ { i } \right|$. We may use without proof the fact that these define norms on the vector space $\mathbb { C } ^ { d }$.
If $A$ is a matrix in $\mathrm { M } _ { d } ( \mathbb { C } )$ we denote by $\operatorname { Sp } ( A )$ the spectrum of $A$ and we define the spectral radius $\sigma ( A )$ by
$$\sigma ( A ) = \max \{ | \lambda | , \lambda \in \operatorname { Sp } ( A ) \} .$$

Part 1: Adapted norms

1. Let $A \in \mathrm { M } _ { d } ( \mathbb { C } )$. Determine a necessary and sufficient condition on $A$ for the map $x \mapsto \| A x \| _ { \infty }$ to define a norm on $\mathbb { C } ^ { d }$.
2. Given a matrix $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ we define

$$\| A \| = \sup _ { \| x \| _ { \infty } \leqslant 1 } \| A x \| _ { \infty } .$$
a. Show that this defines a norm on $\mathrm { M } _ { d } ( \mathbb { C } )$ and that there exists $x _ { 0 } \in \mathbb { C } ^ { d }$ such that $\left\| x _ { 0 } \right\| _ { \infty } = 1$ and $\left\| A x _ { 0 } \right\| _ { \infty } = \| A \|$. b. Show that for all $( A , B ) \in \mathrm { M } _ { d } ( \mathbb { C } )$ we have $\| A B \| \leqslant \| A \| \cdot \| B \|$.
3. For $1 \leqslant i \leqslant d$ we define $L _ { i } = \left( a _ { i , j } \right) _ { 1 \leqslant j \leqslant d }$ as the $i ^ { \mathrm { th } }$ row vector of $A$. Show that
$$\| A \| = \max _ { 1 \leqslant i \leqslant d } \left\| L _ { i } \right\| _ { 1 } .$$

1. a. Let $u \in \mathcal { L } \left( \mathbb { C } ^ { d } \right)$ be an endomorphism of $\mathbb { C } ^ { d }$ and $M = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ the matrix of $u$ in a basis $\mathcal { B } = \left( e _ { 1 } , \ldots , e _ { d } \right)$. Express the matrix $M ^ { \prime } = \left( m _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant d }$ of $u$ in the basis $\mathcal { B } ^ { \prime } = \left( \alpha _ { 1 } e _ { 1 } , \ldots , \alpha _ { d } e _ { d } \right)$, where the $\alpha _ { i }$ are complex numbers. b. Suppose that $M$ is upper triangular. Show that for all $\varepsilon > 0$ we can choose the $\alpha _ { i }$ such that for $j > i$ we have $\left| m _ { i , j } ^ { \prime } \right| < \varepsilon$.
2. Let $T = \left( t _ { i , j } \right) _ { 1 \leqslant i , j \leqslant d }$ be an upper triangular matrix. Show that for all $\varepsilon > 0$, there exists a norm $\| \cdot \| ^ { \prime }$ on $\mathbb { C } ^ { d }$ such that for all $x \in \mathbb { C } ^ { d }$ we have

$$\| T x \| ^ { \prime } \leqslant ( \sigma ( T ) + \varepsilon ) \| x \| ^ { \prime }$$
(you may choose $\| \cdot \| ^ { \prime }$ in the form $\| x \| ^ { \prime } = \| P x \| _ { \infty }$ for a suitably chosen matrix $P$)

124- What is $\displaystyle\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \dfrac{1 + \cos 2x}{2\sin^2 x}\, dx$?
(1) $1 - \sqrt{2}$ (2) $1 - \dfrac{\pi}{4}$ (3) $\dfrac{\pi}{2} - 1$ (4) $\dfrac{2}{4}$
%% Page 6 Mathematics 120-C Page 5

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

23. $\int \sqcap / 33 \pi / 4 \mathrm { dx } / ( 1 + \cos \mathrm { x } )$ is equal to :
(A) 2
(B) - 2
(C) $1 / 2$
(D) $- 1 / 2$

26. Let $f ( x ) = \int ^ { x } { } _ { 1 } \sqrt { } \left( 2 - t ^ { 2 } \right)$ The real roots of the equation $x ^ { 2 } - f ^ { \prime } ( x ) = 0$ are
(A) $\quad \pm 1$
(B) $\pm 1 / \sqrt { } 2$
(C) $\quad \pm 1 / 2$
(D) 0 and 1

7. If $\int \sin x ^ { 1 } t ^ { 2 } f ( t ) d t = 1 - \sin x \forall x \hat { I } [ 0 , \Pi / 2 ]$ then $f ( 1 / \sqrt { } 3 )$ is:
(a) 3
(b) $\sqrt { } 3$
(c) $1 / 3$
(d) none of these

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
$\int _ { - 1 } ^ { 1 } g ^ { \prime } ( x ) d x =$
(A) $2 g ( - 1 )$
(B) 0
(C) $- 2 g ( 1 )$
(D) $2 g ( 1 )$

The value of the integral $$\int_0^{\pi/2} \frac{3\sqrt{\cos\theta}}{(\sqrt{\cos\theta} + \sqrt{\sin\theta})^5}\,d\theta$$ equals

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be the function defined as $f ( x ) = \sqrt { n }$ if $x \in \left[ \frac { 1 } { n + 1 } , \frac { 1 } { n } \right)$ where $n \in \mathbb { N }$. Let $g : ( 0,1 ) \rightarrow \mathbb { R }$ be a function such that $\int _ { x ^ { 2 } } ^ { x } \sqrt { \frac { 1 - t } { t } } d t < g ( x ) < 2 \sqrt { x }$ for all $x \in ( 0,1 )$. Then $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3

$\int \frac { \sin ^ { 8 } x - \cos ^ { 8 } x } { \left( 1 - 2 \sin ^ { 2 } x \cos ^ { 2 } x \right) } d x$ is equal to
(1) $- \frac { 1 } { 2 } \sin 2 x + c$
(2) $- \sin ^ { 2 } x + c$
(3) $- \frac { 1 } { 2 } \sin x + c$
(4) $\frac { 1 } { 2 } \sin 2 x + c$

Let $I_n = \int \tan^n x\, dx$ $(n > 1)$. If $I_4 + I_6 = a\tan^5 x + bx^5 + c$, then the ordered pair $(a, b)$ is equal to
(1) $\left(-\dfrac{1}{5}, 1\right)$
(2) $\left(\dfrac{1}{5}, 0\right)$
(3) $\left(\dfrac{1}{5}, -1\right)$
(4) $\left(-\dfrac{1}{5}, 0\right)$

The integral $\int \sqrt { 1 + 2 \cot x ( \operatorname { cosec } x + \cot x ) } d x , \left( 0 < x < \frac { \pi } { 2 } \right)$ is equal to
(1) $2 \log \left| \sin \frac { x } { 2 } \right| + c$
(2) $4 \log \left| \sin \frac { x } { 2 } \right| + c$
(3) $4 \log \left| \cos \frac { x } { 2 } \right| + c$
(4) $2 \log \left| \cos \frac { x } { 2 } \right| + c$

The integral $\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 4 } } \frac { 8 \cos 2 x } { ( \tan x + \cot x ) ^ { 3 } } d x$ equals
(1) $\frac { 13 } { 256 }$
(2) $\frac { 15 } { 64 }$
(3) $\frac { 13 } { 32 }$
(4) $\frac { 15 } { 128 }$