Found problems: 1687
2005 Today's Calculation Of Integral, 36
A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$,
\[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\]
Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$
Evaluate
\[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]
2003 China Second Round Olympiad, 2
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.
2023 CMIMC Integration Bee, 1
\[\int_2^0 x^2+3\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2012 Today's Calculation Of Integral, 776
Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$
2003 VJIMC, Problem 4
Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that
$$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$
1951 Miklós Schweitzer, 16
Let $ \mathcal{F}$ be a surface which is simply covered by two systems of geodesics such that any two lines belonging to different systems form angles of the same opening. Prove that $ \mathcal{F}$ can be developed (that is, isometrically mapped) into the plane.
2010 Contests, 1
Let $0 < a < b$. Prove that
$\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.
2011 Today's Calculation Of Integral, 675
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
[i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]
2024 CMI B.Sc. Entrance Exam, 2
$g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt$
(a) Find the domain of $g(x)$
(b) Approximate the value of $g(1000)$
(c) Find $x \in [10, 1000]$ to maximize the slope of $g(x)$
(d) Find $x \in [10, 1000]$ to minimize the slope of $g(x)$
(e) Determine, if it exists, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$
2010 Today's Calculation Of Integral, 603
Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$.
Please solve the problem without using partial differentiation for those who don't learn it.
1961 Waseda University entrance exam/Science and Technology
2005 Today's Calculation Of Integral, 24
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.
2005 Putnam, A5
Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$
2010 Today's Calculation Of Integral, 566
In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.
2019 Jozsef Wildt International Math Competition, W. 12
If $0 < a < b$ then: $$\frac{\int \limits^{\frac{a+b}{2}}_{a}\left(\tan^{-1}t\right)dt}{\int \limits_{a}^{b}\left(\tan^{-1}t\right)dt}<\frac{1}{2}$$
2015 VTRMC, Problem 5
Evaluate $\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx$ (where $0\le\operatorname{arctan}(x)<\frac\pi2$ for $0\le x<\infty$).
2013 Today's Calculation Of Integral, 871
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
2005 Today's Calculation Of Integral, 80
Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$.
Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$
about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.
2010 Today's Calculation Of Integral, 526
For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?
1991 Arnold's Trivium, 60
Is there a solution of the Cauchy problem
\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]
in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?
2013 Today's Calculation Of Integral, 899
Find the limit as below.
\[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]
2007 Harvard-MIT Mathematics Tournament, 3
Let $a$ be a positive real number. Find the value of $a$ such that the definite integral \[\int_a^{a^2} \dfrac{dx}{x+\sqrt{x}}\] achieves its smallest possible value.
2009 Today's Calculation Of Integral, 467
Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$.
(1) Find the maximum value of $ y$.
(2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis.
(3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.
2019 Jozsef Wildt International Math Competition, W. 21
Let $f$ be a continuously differentiable function on $[0, 1]$ and $m \in \mathbb{N}$. Let $A = f(1)$ and let $B=\int \limits_{0}^1 x^{-\frac{1}{m}}f(x)dx$. Calculate $$\lim \limits_{n \to \infty} n\left(\int \limits_{0}^1 f(x)dx-\sum \limits_{k=1}^n \left(\frac{k^m}{n^m}-\frac{(k-1)^m}{n^m}\right)f\left(\frac{(k-1)^m}{n^m}\right)\right)$$in terms of $A$ and $B$.
2011 Today's Calculation Of Integral, 728
Evaluate
\[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]
2010 Today's Calculation Of Integral, 664
For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$.
Find $I_1+I_2+I_3+I_4$.
[i]1992 University of Fukui entrance exam/Medicine[/i]