Directly evaluate a definite integral of an explicitly given function, possibly involving symmetry properties, special techniques, or parameter determination to find a numerical or symbolic result.
Statements (13) $\lim _ { x \rightarrow 0 } e ^ { \frac { 1 } { x } } = + \infty$. (14) The following inequality is true. $$\lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { 100 } } < \lim _ { x \rightarrow \infty } \frac { \ln x } { x ^ { \frac { 1 } { 100 } } }$$ (15) For any positive integer $n$, $$\int _ { - n } ^ { n } x ^ { 2023 } \cos ( n x ) \, dx < \frac { n } { 2023 }$$ (16) There is no polynomial $p ( x )$ for which there is a single line that is tangent to the graph of $p ( x )$ at exactly 100 points.
The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$. $$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$ Find the value of $a ^ { 4 }$. [3 points]
When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points] (1) $- 4$ (2) $- 2$ (3) 0 (4) 2 (5) 4
A quadratic function $f ( x )$ satisfies $f ( 0 ) = - 1$ and $$\int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x$$ What is the value of $f ( 2 )$? [4 points] (1) 11 (2) 10 (3) 9 (4) 8 (5) 7
A quadratic function $f ( x )$ satisfies $f ( 0 ) = 0$ and the following conditions. (가) $\int _ { 0 } ^ { 2 } | f ( x ) | d x = - \int _ { 0 } ^ { 2 } f ( x ) d x = 4$ (나) $\int _ { 2 } ^ { 3 } | f ( x ) | d x = \int _ { 2 } ^ { 3 } f ( x ) d x$ Find the value of $f ( 5 )$. [4 points]
What is the area of the region enclosed by the curve $y = e ^ { 2 x }$, the $x$-axis, and the two lines $x = \ln \frac { 1 } { 2 }$ and $x = \ln 2$? [3 points] (1) $\frac { 5 } { 3 }$ (2) $\frac { 15 } { 8 }$ (3) $\frac { 15 } { 7 }$ (4) $\frac { 5 } { 2 }$ (5) 3
A function $f ( x )$ differentiable on the entire set of real numbers satisfies the following conditions. (a) On the closed interval $[ 0,1 ]$, $f ( x ) = x$. (b) For some constants $a , b$, on the interval $[ 0 , \infty )$, $f ( x + 1 ) - x f ( x ) = a x + b$. Find the value of $60 \times \int _ { 1 } ^ { 2 } f ( x ) d x$. [4 points]
A cubic function $f(x)$ satisfies $$xf(x) - f(x) = 3x^4 - 3x$$ for all real numbers $x$. Find the value of $\int_{-2}^{2} f(x)\,dx$. [3 points] (1) 12 (2) 16 (3) 20 (4) 24 (5) 28
For the function $f(x) = 3x^{2} - 16x - 20$, $$\int_{-2}^{a} f(x)\, dx = \int_{-2}^{0} f(x)\, dx$$ When this condition is satisfied, what is the value of the positive number $a$? [4 points] (1) 16 (2) 14 (3) 12 (4) 10 (5) 8
As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points] (1) $\ln(e+1)$ (2) $\ln(e+2)$ (3) $\ln(e+3)$ (4) $\ln(2e+1)$ (5) $\ln(2e+2)$
We consider a rectangle $]a,b[ \times ]c,d[$ of the plane $\mathbb{R}^{2}$, with $a < b$ and $c < d$. Calculate the real number $V(]a,b[ \times ]c,d[)$. What does it represent? (One may use functions of the type $$(x,y) \mapsto f(x,y) = \phi(x)\varphi(y)$$ where $\phi$ and $\varphi$ are well-chosen continuous and piecewise affine functions).
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: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$ II.D.1) For all $n \in \mathbb{N}$, we set $I_n = \int_0^1 P_n(u) \, du$. Calculate, for all $n \in \mathbb{N}$, the value of $I_n$. II.D.2) Deduce for all $n \in \mathbb{N}$ the relation: $\langle L_n, L_n \rangle = \frac{1}{2n+1}$.
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the unique family of polynomials such that for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$. Calculate $K_0$, $K_1$ and $K_2$.
For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm. Let $n \in \mathbb{N}$. Show that there exists a unique polynomial $\Pi_n \in \mathbb{R}_n[X]$ such that $$\left\|\Pi_n - f\right\| = \min_{Q \in \mathbb{R}_n[X]} \|Q - f\|$$