Found problems: 1687
2005 Today's Calculation Of Integral, 86
Prove
\[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]
2005 Today's Calculation Of Integral, 70
Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$
2013 Romania National Olympiad, 1
Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .
2011 Today's Calculation Of Integral, 729
Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$
2007 Today's Calculation Of Integral, 180
Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$
Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$
1987 Vietnam National Olympiad, 2
Let $ f : [0, \plus{}\infty) \to \mathbb R$ be a differentiable function. Suppose that $ \left|f(x)\right| \le 5$ and $ f(x)f'(x) \ge \sin x$ for all $ x \ge 0$. Prove that there exists $ \lim_{x\to\plus{}\infty}f(x)$.
2012 Today's Calculation Of Integral, 815
Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$
2012 Online Math Open Problems, 25
Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$, and then written in nondecreasing order as $x_1\le x_2\le\cdots\le x_{2012}$. If the probability that $x_{i+1} - x_i \le \frac{1}{2011}$ for $i=1,2,\ldots,2011$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$, find the remainder when $m+n$ is divided by 1000.
[i]Victor Wang.[/i]
2010 Today's Calculation Of Integral, 542
Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.
2002 Romania National Olympiad, 2
Let $f:[0,1]\rightarrow\mathbb{R}$ be an integrable function such that:
\[0<\left\vert \int_{0}^{1}f(x)\, \text{d}x\right\vert\le 1.\]
Show that there exists $x_1\not= x_2, x_1,x_2\in [0,1]$, such that:
\[\int_{x_1}^{x_2}f(x)\, \text{d}x=(x_1-x_2)^{2002}\]
2009 Today's Calculation Of Integral, 462
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.
2010 Today's Calculation Of Integral, 545
(1) Evaluate $ \int_0^1 xe^{x^2}dx$.
(2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.
2008 Moldova MO 11-12, 6
Find $ \lim_{n\to\infty}a_n$ where $ (a_n)_{n\ge1}$ is defined by $ a_n\equal{}\frac1{\sqrt{n^2\plus{}8n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}16n\minus{}1}}\plus{}\frac1{\sqrt{n^2\plus{}24n\minus{}1}}\plus{}\ldots\plus{}\frac1{\sqrt{9n^2\minus{}1}}$.
2010 Today's Calculation Of Integral, 574
Let $ n$ be a positive integer. Prove that $ x^ne^{1\minus{}x}\leq n!$ for $ x\geq 0$,
2012 Today's Calculation Of Integral, 795
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$
2012 Today's Calculation Of Integral, 806
Let $n$ be positive integers and $t$ be a positive real number.
Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$
2009 Today's Calculation Of Integral, 466
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$.
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$.
(2) Find $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$.
2012 Today's Calculation Of Integral, 789
Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$
2013 Today's Calculation Of Integral, 886
Find the functions $f(x),\ g(x)$ such that
$f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$
$g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$
2024 SEEMOUS, P3
For every $n\geq 1$ define $x_n$ by $$x_n=\int_0^1 \ln(1+x+x^2+\dots +x^n)\cdot\ln\frac{1}{1-x}\mathrm dx.$$
a) Show that $x_n$ is finite for every $n\geq 1$ and $\lim_{n\rightarrow\infty}x_n=2$.
b) Calculate $\lim_{n\rightarrow\infty}\frac{n}{\ln n}(2-x_n)$.
1952 AMC 12/AHSME, 47
In the set of equations $ z^x \equal{} y^{2x}, 2^z \equal{} 2\cdot4^x, x \plus{} y \plus{} z \equal{} 16$, the integral roots in the order $ x,y,z$ are:
$ \textbf{(A)}\ 3,4,9 \qquad\textbf{(B)}\ 9, \minus{} 5 \minus{} ,12 \qquad\textbf{(C)}\ 12, \minus{} 5,9 \qquad\textbf{(D)}\ 4,3,9 \qquad\textbf{(E)}\ 4,9,3$
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
1980 Putnam, A3
Evaluate
$$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$
2005 Rioplatense Mathematical Olympiad, Level 3, 3
Let $k$ be a positive integer. Show that for all $n>k$ there exist convex figures $F_{1},\ldots, F_{n}$ and $F$ such that there doesn't exist a subset of $k$ elements from $F_{1},..., F_{n}$ and $F$ is covered for this elements, but $F$ is covered for every subset of $k+1$ elements from $F_{1}, F_{2},....., F_{n}$.
1971 Canada National Olympiad, 5
Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.