Found problems: 1687
2010 Today's Calculation Of Integral, 609
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number.
1972 Tokyo University of Education entrance exam
1969 AMC 12/AHSME, 19
The number of distinct ordered pairs $(x,y)$, where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$, is:
$\textbf{(A) }0\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }12\qquad
\textbf{(E) }\text{infinite}$
2012 Today's Calculation Of Integral, 848
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
1986 Traian Lălescu, 2.1
Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds:
$$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$
2009 Putnam, A6
Let $ f: [0,1]^2\to\mathbb{R}$ be a continuous function on the closed unit square such that $ \frac{\partial f}{\partial x}$ and $ \frac{\partial f}{\partial y}$ exist and are continuous on the interior of $ (0,1)^2.$ Let $ a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx$ and $ d\equal{}\int_0^1f(x,1)\,dx.$ Prove or disprove: There must be a point $ (x_0,y_0)$ in $ (0,1)^2$ such that
$ \frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a$ and $ \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c.$
2012 Today's Calculation Of Integral, 839
Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$
2024 CMIMC Integration Bee, 13
\[\int_0^{2\pi} \frac{1}{3+2 \sqrt{3} \cos x + \cos^2 x}\mathrm dx\]
[i]Proposed by Robert Trosten[/i]
Today's calculation of integrals, 881
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
1973 Putnam, B4
(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that
$$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$
(b) Show an example in which equality occurs.
2009 Today's Calculation Of Integral, 442
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$
2003 Romania National Olympiad, 2
Let be an odd natural number $ n\ge 3. $ Find all continuous functions $ f:[0,1]\longrightarrow\mathbb{R} $ that satisfy the following equalities.
$$ \int_0^1 \left( f\left(\sqrt[k]{x}\right) \right)^{n-k} dx=k/n,\quad\forall k\in\{ 1,2,\ldots ,n-1\} $$
[i]Titu Andreescu[/i]
2015 Kyoto University Entry Examination, 5
5. Let $a,b,c,d,e$ be positive rational numbers. Consider integral expressions
$f(x)=ax^2+bx+c$
$g(x)=dx+e$
Put $\frac{f(n)}{g(n)}$ an integer for all positive integers $n$. Then, show that $f(x)$ is dividible by $g(x)$.
2013 Kosovo National Mathematical Olympiad, 2
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
2011 Today's Calculation Of Integral, 688
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
2021 CMIMC Integration Bee, 9
$$\int_1^2\frac{12x^3+12x+12}{2x^4+3x^2+4x}\,dx$$
[i]Proposed by Connor Gordon[/i]
2007 Today's Calculation Of Integral, 193
For $a>0$, let $l$ be the line created by rotating the tangent line to parabola $y=x^{2}$, which is tangent at point $A(a,a^{2})$, around $A$ by $-\frac{\pi}{6}$.
Let $B$ be the other intersection of $l$ and $y=x^{2}$. Also, let $C$ be $(a,0)$ and let $O$ be the origin.
(1) Find the equation of $l$.
(2) Let $S(a)$ be the area of the region bounded by $OC$, $CA$ and $y=x^{2}$. Let $T(a)$ be the area of the region bounded by $AB$ and $y=x^{2}$. Find $\lim_{a \to \infty}\frac{T(a)}{S(a)}$.
PEN Q Problems, 9
For non-negative integers $n$ and $k$, let $P_{n, k}(x)$ denote the rational function \[\frac{(x^{n}-1)(x^{n}-x) \cdots (x^{n}-x^{k-1})}{(x^{k}-1)(x^{k}-x) \cdots (x^{k}-x^{k-1})}.\] Show that $P_{n, k}(x)$ is actually a polynomial for all $n, k \in \mathbb{N}$.
1981 Miklós Schweitzer, 6
Let $ f$ be a strictly increasing, continuous function mapping $ I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $ x,y \in I$: \[ 1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .\]
[i]Zs. Pales[/i]
1990 AMC 12/AHSME, 7
A triangle with integral sides has perimeter $8$. The area of the triangle is
$\textbf{(A) }2\sqrt{2}\qquad
\textbf{(B) }\dfrac{16}{9}\sqrt{3}\qquad
\textbf{(C) }2\sqrt{3}\qquad
\textbf{(D) }4\qquad
\textbf{(E) }4\sqrt{2}$
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 4
Let $\alpha,\ \beta$ be real numbers. Find the ranges of $\alpha,\ \beta$ such that the improper integral $\int_1^{\infty} \frac{x^{\alpha}\ln x}{(1+x)^{\beta}}$ converges.
1987 AIME Problems, 11
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.
2005 Today's Calculation Of Integral, 26
Evaluate
\[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]
2009 Today's Calculation Of Integral, 517
Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.
1980 Putnam, A3
Evaluate
$$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$
2005 Today's Calculation Of Integral, 22
Evaluate
\[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]