Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$.
Show that $E$ is an $\mathbb{R}$-vector space.

Let $E$ be the set of continuous functions $f$ from $I$ to $\mathbb{R}$ such that $f^2 w$ is integrable on $I$. For all functions $f$ and $g$ in $E$, we set $$\langle f, g \rangle = \int_I f(x) g(x) w(x)\,\mathrm{d}x.$$
Show that we thus define an inner product on $E$.

For any real $\alpha > 0$, consider $J _ { \alpha } = 2 \ln ( 2 ) - \ln \left( 1 + \alpha ^ { 2 } \right) - 2 \alpha \arctan \left( \frac { 1 } { \alpha } \right)$. Show that there exists $\gamma > 0$ such that, for all $\alpha \in ] 0 , \gamma [$, $J _ { \alpha } > 0$.

For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n }$ given, for $k \in \llbracket 0 , n - 1 \rrbracket$, by $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ and $$S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right).$$ For all $n \in \mathbb { N } ^ { * }$, show that $$\int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t + \frac { 1 } { n } h _ { \alpha } \left( \frac { 2 n - 1 } { 2 n } \right) \leqslant S _ { n } \left( h _ { \alpha } \right) \leqslant \frac { 1 } { n } h _ { \alpha } \left( \frac { 1 } { 2 n } \right) + \int _ { 1 / 2 n } ^ { ( 2 n - 1 ) / 2 n } h _ { \alpha } ( t ) \, \mathrm { d } t.$$

For all $n \in \mathbb { N } ^ { * }$, consider in $]0,1[$ the points $a _ { k , n } = \frac { 2 k + 1 } { 2 n }$ for $k \in \llbracket 0 , n - 1 \rrbracket$, and $S _ { n } \left( h _ { \alpha } \right) = \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } h _ { \alpha } \left( a _ { k , n } \right)$, where $J_\alpha = \int_0^1 h_\alpha(t)\,\mathrm{d}t$. Deduce that the sequence $\left( S _ { n } \left( h _ { \alpha } \right) \right) _ { n \in \mathbb { N } ^ { * } }$ converges to $J _ { \alpha }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
For $k \geqslant 1$, show that if $\varphi \in \mathcal{C}_{c}^{k+1}(\mathbb{R})$, we have
$$\left\|(T_{\mu}\varphi)^{(k)} - \varphi^{(k)}\right\|_{\infty} \leqslant \frac{\mu}{2} \left\|\varphi^{(k+1)}\right\|_{\infty}.$$

Show that $$I _ { n } \geqslant \frac { 1 } { 2 ^ { n } }.$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Let $I$ be an interval of $\mathbb { R }$ and $\left( f _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ a sequence of functions piecewise continuous on $I$ that converges uniformly on $I$ to a function $f$ also piecewise continuous on $I$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ (respectively $\left( v _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$) is a sequence of real numbers belonging to $I$ that converges to $u \in I$ (respectively $v \in I$), show that $$\lim _ { n \rightarrow + \infty } \left( \int _ { u _ { n } } ^ { v _ { n } } f _ { n } ( x ) \mathrm { d } x \right) = \int _ { u } ^ { v } f ( x ) \mathrm { d } x$$

Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a piecewise continuous function taking values in an interval $J$. Let $\varphi$ be a continuous and convex function on $J$. Prove that
$$\varphi \left( \frac { 1 } { b - a } \int _ { a } ^ { b } f ( t ) \mathrm { d } t \right) \leqslant \frac { 1 } { b - a } \int _ { a } ^ { b } \varphi \circ f ( t ) \mathrm { d } t .$$
You may use Riemann sums.

Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Prove that, for all $x > 0$,
$$\exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \leqslant \frac { \mathrm { e } } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t$$
You may note that $\ln ( f ( t ) ) = \ln ( t f ( t ) ) - \ln ( t )$.

Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Deduce that $x \mapsto \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right)$ is integrable on $\mathbb { R } _ { + } ^ { * }$ and that
$$\int _ { 0 } ^ { + \infty } \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \mathrm { d } x \leqslant \mathrm { e } \int _ { 0 } ^ { + \infty } f ( x ) \mathrm { d } x$$

Let $f \in \mathcal { C } ^ { 0 } ( [ 0 , + \infty [ )$ and $\ell \in \mathbb { R }$. Prove that $$\left( \lim _ { x \rightarrow + \infty } f ( x ) = \ell \right) \Rightarrow \left( \lim _ { x \rightarrow + \infty } \frac { 1 } { x } \int _ { 0 } ^ { x } f ( t ) d t = \ell \right)$$

Using a counterexample, prove that the converse of the result in question 3.1 is false, i.e. that $$\left( \lim _ { x \rightarrow + \infty } \frac { 1 } { x } \int _ { 0 } ^ { x } f ( t ) d t = \ell \right) \not\Rightarrow \left( \lim _ { x \rightarrow + \infty } f ( x ) = \ell \right)$$ for $f \in \mathcal{C}^0([0,+\infty[)$.

We fix $( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 }$ and set $\alpha _ { p , q } := \dfrac { p } { q }$. We define, for all $t \in \mathbf { R } _ { + }$, the application $I _ { p , q } : \mathbf { R } _ { + } \rightarrow \mathbf { R }$ by $$I _ { p , q } ( t ) := \int _ { 0 } ^ { 1 } \frac { x ^ { ( t + 1 ) \alpha _ { p , q } } } { 1 + x ^ { \alpha _ { p , q } } } d x$$ Determine $$\lim _ { n \rightarrow + \infty } I _ { p , q } ( n ) = 0$$

Let $f$ be a differentiable function on $[0, 2\pi]$ with $f'(x)$ increasing. Show that $\int_0^{2\pi} f(x) \cos x \, dx \geq 0$.

Let $f ( x ) = ( \tan x ) ^ { 3 / 2 } - 3 \tan x + \sqrt{\tan x}$. Consider the three integrals $I _ { 1 } = \int_0^1 f ( x ) \, dx$; $I _ { 2 } = \int_{0.3}^{1.3} f ( x ) \, dx$ and $I _ { 3 } = \int_{0.5}^{1.5} f ( x ) \, dx$. Then,
(a) $I _ { 1 } > I _ { 2 } > I _ { 3 }$
(b) $I _ { 2 } > I _ { 1 } > I _ { 3 }$
(c) $I _ { 3 } > I _ { 1 } > I _ { 2 }$
(d) $I _ { 1 } > I _ { 3 } > I _ { 2 }$

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3.

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $(2, 3)$
(D) is greater than 3

The value of the integral $$\int _ { 2 } ^ { 3 } \frac { d x } { \log _ { e } x }$$ (A) is less than 2
(B) is equal to 2
(C) lies in the interval $( 2, 3 )$
(D) is greater than 3

Let $I = \int_0^1 \frac{\sin x}{\sqrt{x}}\,dx$ and $J = \int_0^1 \frac{\cos x}{\sqrt{x}}\,dx$. Then which one of the following is true?
(A) $I > \frac{2}{3}$ and $J > 2$
(B) $I < \frac{2}{3}$ and $J < 2$
(C) $I < \frac{2}{3}$ and $J > 2$
(D) $I > \frac{2}{3}$ and $J < 2$

Let $f : \left[ \frac { 1 } { 2 } , 1 \right] \rightarrow \mathbb { R }$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f ^ { \prime } ( x ) < 2 f ( x )$ and $f \left( \frac { 1 } { 2 } \right) = 1$. Then the value of $\int _ { 1/2 } ^ { 1 } f ( x ) d x$ lies in the interval
(A) $( 2 e - 1,2 e )$
(B) $( e - 1,2 e - 1 )$
(C) $\left( \frac { e - 1 } { 2 } , e - 1 \right)$
(D) $\left( 0 , \frac { e - 1 } { 2 } \right)$

Let $f ^ { \prime } ( x ) = \frac { 192 x ^ { 3 } } { 2 + \sin ^ { 4 } \pi x }$ for all $x \in \mathbb { R }$ with $f \left( \frac { 1 } { 2 } \right) = 0$. If $m \leq \int _ { 1 / 2 } ^ { 1 } f ( x ) d x \leq M$, then the possible values of $m$ and $M$ are
(A) $m = 13 , M = 24$
(B) $\quad m = \frac { 1 } { 4 } , M = \frac { 1 } { 2 }$
(C) $m = - 11 , M = 0$
(D) $m = 1 , M = 12$

Let $f : \mathbb{R} \rightarrow (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval $(0,1)$?
[A] $x^9 - f(x)$
[B] $x - \int_0^{\frac{\pi}{2} - x} f(t)\cos t\, dt$
[C] $e^x - \int_0^x f(t)\sin t\, dt$
[D] $f(x) + \int_0^{\frac{\pi}{2}} f(t)\sin t\, dt$

Which of the following inequalities is/are TRUE?
(A) $\int _ { 0 } ^ { 1 } x \cos x \, d x \geq \frac { 3 } { 8 }$
(B) $\int _ { 0 } ^ { 1 } x \sin x \, d x \geq \frac { 3 } { 10 }$
(C) $\int _ { 0 } ^ { 1 } x ^ { 2 } \cos x \, d x \geq \frac { 1 } { 2 }$
(D) $\int _ { 0 } ^ { 1 } x ^ { 2 } \sin x \, d x \geq \frac { 2 } { 9 }$

Let $f : \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] \rightarrow \mathbb { R }$ be a continuous function such that $$f ( 0 ) = 1 \text { and } \int _ { 0 } ^ { \frac { \pi } { 3 } } f ( t ) d t = 0$$ Then which of the following statements is (are) TRUE ?
(A) The equation $f ( x ) - 3 \cos 3 x = 0$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(B) The equation $f ( x ) - 3 \sin 3 x = - \frac { 6 } { \pi }$ has at least one solution in $\left( 0 , \frac { \pi } { 3 } \right)$
(C) $\lim _ { x \rightarrow 0 } \frac { x \int _ { 0 } ^ { x } f ( t ) d t } { 1 - e ^ { x ^ { 2 } } } = - 1$
(D) $\lim _ { x \rightarrow 0 } \frac { \sin x \int _ { 0 } ^ { x } f ( t ) d t } { x ^ { 2 } } = - 1$