LFM Pure

grandes-ecoles 2011 QIV.C Prove an Integral Inequality or Bound View

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: $$\int_{0}^{+\infty} \frac{h(u)}{u+x} du$$

grandes-ecoles 2011 QV.A Prove an Integral Identity or Equality View

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Show that for all natural integers $i$: $$\int_{x+i}^{x+i+1} \ln t \, dt = \ln(x+i) - \int_{i}^{i+1} \frac{u-i-1}{u+x} du$$

grandes-ecoles 2011 QII.C Inner Product or Orthogonality Proof via Integration by Parts View

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
Let $n \in \mathbb{N}^*$. Show that, for all $Q \in \mathbb{R}_{n-1}[X]$, $\langle Q, L_n \rangle = 0$.
Hint: you may integrate by parts.

grandes-ecoles 2012 QVIII.D Inner Product or Orthogonality Proof via Integration by Parts View

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Show that for all $P$ and $Q$ in $\mathcal{P}$, we have $$\langle U(P), Q \rangle = \langle P, U(Q) \rangle.$$

grandes-ecoles 2015 QII.B.2 Prove an Integral Identity or Equality View

If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$. If $f$ is of class $\mathcal{C}^1$, show that $(T_f)' = T_{f'}$. Adapt this result to the case where $f$ is piecewise of class $\mathcal{C}^1$.

grandes-ecoles 2015 QI.D.2 Reduction Formula or Recurrence via Integration by Parts View

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Show that $\left. \forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \forall \varepsilon \in \right] 0,1 \left[ , \quad I _ { p , q } ^ { \varepsilon } = - \frac { q } { p + 1 } I _ { p , q - 1 } ^ { \varepsilon } - \frac { \varepsilon ^ { p + 1 } ( \ln \varepsilon ) ^ { q } } { p + 1 } \right.$.

grandes-ecoles 2015 QI.D.3 Reduction Formula or Recurrence via Integration by Parts View

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce that we have $\forall p \in \mathbb { N } , \forall q \in \mathbb { N } ^ { * } , \quad I _ { p , q } = - \frac { q } { p + 1 } I _ { p , q - 1 }$.

grandes-ecoles 2015 QI.D.4 Reduction Formula or Recurrence via Integration by Parts View

For every pair of natural integers $( p , q )$ and for every $\varepsilon \in ] 0,1 [$, we denote $$I _ { p , q } = \int _ { 0 } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t \quad \text { and } \quad I _ { p , q } ^ { \varepsilon } = \int _ { \varepsilon } ^ { 1 } t ^ { p } ( \ln t ) ^ { q } \mathrm {~d} t$$
Deduce an expression for $I _ { p , q }$ as a function of the integers $p$ and $q$.

grandes-ecoles 2015 QI.E Prove an Integral Identity or Equality View

Let $r$ be a non-zero natural integer and $f$ a function expandable as a power series on $] - 1,1 [$. We assume that for every $x$ in $] - 1,1 \left[ , f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } \right.$ and that $\sum _ { n \geqslant 0 } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$ converges absolutely.
Show that $\int _ { 0 } ^ { 1 } ( \ln t ) ^ { r - 1 } f ( t ) \mathrm { d } t = ( - 1 ) ^ { r - 1 } ( r - 1 ) ! \sum _ { n = 0 } ^ { + \infty } \frac { a _ { n } } { ( n + 1 ) ^ { r } }$.

grandes-ecoles 2016 QIV.E Prove an Integral Inequality or Bound View

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
We henceforth admit that $g$ is of class $C^{1}$ on $[-1,1]$.
Using integration by parts, show the existence of a real number $C$ such that
$$\forall n \in \mathbb{N}, \quad \left|\int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x\right| \leqslant \frac{C}{2n+1}$$

grandes-ecoles 2020 Q4 Prove an Integral Inequality or Bound View

For $x \in [ 2 , + \infty[$, we set $$I ( x ) = \int _ { 2 } ^ { x } \frac { \mathrm{dt} } { \ln ( t ) }$$ Justify, for $x \in [ 2 , + \infty[$, the relation $$I ( x ) = \frac { x } { \ln ( x ) } - \frac { 2 } { \ln ( 2 ) } + \int _ { 2 } ^ { x } \frac { \mathrm{dt} } { ( \ln ( t ) ) ^ { 2 } }$$ Establish moreover the relation $$\int _ { 2 } ^ { x } \frac { \mathrm{dt} } { ( \ln ( t ) ) ^ { 2 } } \underset { x \rightarrow + \infty } { = } o ( I ( x ) )$$ Deduce finally an equivalent of $I ( x )$ as $x$ tends to $+ \infty$.

grandes-ecoles 2020 Q27 Inner Product or Orthogonality Proof via Integration by Parts View

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$, equipped with the inner product $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$ Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined by $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$

grandes-ecoles 2020 Q27 Inner Product or Orthogonality Proof via Integration by Parts View

In this part, $E_1$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. We denote by $N$ the norm associated with the inner product $(f \mid g) = \int_0^1 f'(t) g'(t) \, \mathrm{d}t$. For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Prove that the pre-Hilbert space $(E_1, (\cdot \mid \cdot))$ is a reproducing kernel Hilbert space and that its reproducing kernel is the application $K$ defined in the previous part.

grandes-ecoles 2020 Q28 Inner Product or Orthogonality Proof via Integration by Parts View

We consider the space $E$ of continuous functions from $[0,1]$ to $\mathbb{R}$, equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ Show that $(E, \langle \cdot, \cdot \rangle)$ is not a reproducing kernel Hilbert space.

grandes-ecoles 2020 Q28 Inner Product or Orthogonality Proof via Integration by Parts View

We consider the space $E$ of continuous functions from $[0,1]$ to $\mathbb{R}$, equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ Show that $(E, \langle \cdot, \cdot \rangle)$ is not a reproducing kernel Hilbert space.

grandes-ecoles 2021 Q19 Reduction Formula or Recurrence via Integration by Parts View

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 $k$, $$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$

grandes-ecoles 2021 Q19 Reduction Formula or Recurrence via Integration by Parts View

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 $k$, $$m_{2k+2} = \frac{2(2k+1)}{k+2} m_{2k}$$

grandes-ecoles 2022 Q16 Integral Involving a Parameter or Operator Identity View

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Show that for all functions $f \in E$ and for all $x > 0$, $$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$

grandes-ecoles 2022 Q16 Integral Involving a Parameter or Operator Identity View

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
Express the integral $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t$$ in terms of $V(f)$.

grandes-ecoles 2022 Q28 Inner Product or Orthogonality Proof via Integration by Parts View

We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$, which is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$. The limits of $t \mapsto F(t)U(g)(t)$ at $0$ and $+\infty$ are both $0$. Show that $$\langle f \mid U ( g ) \rangle = \int _ { 0 } ^ { + \infty } U ( f ) ^ { \prime } ( t ) U ( g ) ^ { \prime } ( t ) \mathrm { e } ^ { - t } \mathrm {~d} t.$$

grandes-ecoles 2022 Q29 Inner Product or Orthogonality Proof via Integration by Parts View

We fix two functions $f$ and $g$ in $E$. Using the result of Question 28, deduce that $\langle f \mid U ( g ) \rangle = \langle U ( f ) \mid g \rangle$.

grandes-ecoles 2023 Q2 Definite Integral Evaluation by Parts View

Find two real numbers $\alpha$ and $\beta$ such that: $$\forall n \in \mathbf{N}^*, \int_0^{\pi} (\alpha t^2 + \beta t) \cos(nt) \mathrm{d}t = \frac{1}{n^2}$$ then verify that if $t \in ]0, \pi]$, then: $$\forall n \in \mathbf{N}^*, \sum_{k=1}^n \cos(kt) = \frac{\sin\left(\frac{(2n+1)t}{2}\right)}{2\sin\left(\frac{t}{2}\right)} - \frac{1}{2}$$

grandes-ecoles 2023 Q3 Prove an Integral Identity or Equality View

Justify that, if $\varphi$ is a $\mathcal{C}^1$ application from $[0, \pi]$ to $\mathbf{R}$, then $$\lim_{x \to +\infty} \int_0^{\pi} \varphi(t) \sin(xt) \mathrm{d}t = 0$$ and conclude that $$\sigma(1) = \frac{\pi^2}{6}$$

grandes-ecoles 2023 Q14 Integral Involving a Parameter or Operator Identity View

For $p \in \mathbb{K}[X]$, express $Jp$ in terms of the derivatives $p^{(k)}$ ($k \in \mathbb{N}$) of $p$, where $J$ is defined by $Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$.

grandes-ecoles 2024 Q16 Prove an Integral Identity or Equality View

Deduce that:
$$\int _ { 0 } ^ { + \infty } ( \cos ( t ) ) ^ { 2 p } \frac { \sin ( t ) } { t } \mathrm {~d} t = \int _ { 0 } ^ { \frac { \pi } { 2 } } ( \cos ( t ) ) ^ { 2 p } \mathrm {~d} t$$
In the case $p = 0$, this integral is commonly called the ``Dirichlet Integral''.

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

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Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

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Standard trigonometric equations 106

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Integration by Substitution 116

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Differential equations 352

3x3 Matrices 144