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 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 $$S _ { p , q } = \frac { ( - 1 ) ^ { \lambda - 1 } } { p } \left( \ln ( 2 ) - \sum _ { k = 1 } ^ { \lambda - 1 } \frac { ( - 1 ) ^ { k - 1 } } { k } \right)$$

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

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

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) $2 \log \left| \cos \frac { x } { 2 } \right| + c$

If $f ( t ) = \int _ { 0 } ^ { \pi } \frac { 2 x \mathrm {~d} x } { 1 - \cos ^ { 2 } \mathrm { t } \sin ^ { 2 } x } , 0 < \mathrm { t } < \pi$, then the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \pi ^ { 2 } \mathrm { dt } } { f ( \mathrm { t } ) }$ equals $\_\_\_\_$

If $\int ( \cos x ) ^ { - 5 / 2 } ( \sin x ) ^ { - 11 / 2 } d x = \frac { p _ { 1 } } { q _ { 1 } } ( \cot x ) ^ { 9 / 2 } + \frac { p _ { 2 } } { q _ { 2 } } ( \cot x ) ^ { 5 / 2 } + \frac { p _ { 3 } } { q _ { 3 } } ( \cot x ) ^ { 1 / 2 } - \frac { p _ { 4 } } { q _ { 4 } } ( \cot x ) ^ { - 3 / 2 } + c$ (where c is constant of integration), then value of $\frac { 15 p _ { 1 } p _ { 2 } p _ { 3 } p _ { 4 } } { q _ { 1 } q _ { 2 } q _ { 3 } q _ { 4 } }$ is
(A) 16
(B) 14

Find the value of
$$\int _ { 1 } ^ { 4 } \frac { 3 - 2 x } { x \sqrt { x } } \mathrm {~d} x$$
A $- \frac { 13 } { 2 }$
B $- \frac { 85 } { 16 }$
C $- \frac { 13 } { 8 }$
D - 1
E $- \frac { 1 } { 4 }$
F $\frac { 7 } { 4 }$
G 7

Consider the two statements R: $\quad k$ is an integer multiple of $\pi$
$$\mathrm { S } : \quad \int _ { 0 } ^ { k } \sin 2 x \mathrm {~d} x = 0$$
Which of the following statements is true? A $R$ is necessary and sufficient for $S$. B R is necessary but not sufficient for $S$. C R is sufficient but not necessary for $S$. D $R$ is not necessary and not sufficient for $S$.

$$\int_{0}^{\frac{\pi}{3}} \frac{\sin x}{\cos^{2} x}\, dx$$
What is the value of the integral?
A) 2
B) 1
C) 0
D) $-1$
E) $-2$

$$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sin 2 x \cdot \cot x \, d x$$
What is the value of this integral?
A) $\frac { \pi + 1 } { 2 }$
B) $\frac { \pi + 1 } { 3 }$
C) $\frac { \pi + 2 } { 4 }$
D) $\frac { \pi - 1 } { 6 }$
E) $\frac { \pi - 2 } { 6 }$