Found problems: 1687
2009 Today's Calculation Of Integral, 411
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.
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
2011 Today's Calculation Of Integral, 699
Find the volume of the part bounded by $z=x+y,\ z=x^2+y^2$ in the $xyz$ space.
2005 Today's Calculation Of Integral, 33
Evaluate
\[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]
2011 Canadian Open Math Challenge, 8
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
1997 IMC, 1
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]
2012 Today's Calculation Of Integral, 823
Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$
(1) Express $y$ in terms of $x$.
(2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$.
(3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.
1972 Swedish Mathematical Competition, 5
Show that
\[
\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}
\]
for all positive integers $n$.
2011 Today's Calculation Of Integral, 732
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
2006 APMO, 2
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
2014 AMC 12/AHSME, 20
For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$?
${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $
2010 Today's Calculation Of Integral, 665
Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$.
[i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]
2002 VJIMC, Problem 4
Prove that
$$\lim_{n\to\infty}n^2\left(\int^1_0\sqrt[n]{1+x^n}\text dx-1\right)=\frac{\pi^2}{12}.$$
2012 Today's Calculation Of Integral, 812
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$
1963 Miklós Schweitzer, 8
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
2012 Today's Calculation Of Integral, 835
Evaluate the following definite integrals.
(a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$
(b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$
(c) $\int_1^e x\ln \sqrt{x}\ dx$
(d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$
2005 Today's Calculation Of Integral, 10
Calculate the following indefinite integrals.
[1] $\int (2x+1)\sqrt{x+2}\ dx$
[2] $\int \frac{1+\cos x}{x+\sin x}\ dx$
[3] $\int \sin ^ 5 x \cos ^ 3 x \ dx$
[4] $\int \frac{(x-3)^2}{x^4}\ dx$
[5] $\int \frac{dx}{\tan x}\ dx$
2011 Today's Calculation Of Integral, 690
Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.
2002 VJIMC, Problem 3
Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying
$$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by
$$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.
2009 Today's Calculation Of Integral, 469
Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.
2014 VJIMC, Problem 4
Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that
$$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$
2009 AIME Problems, 11
Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.
1948 Putnam, B4
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?
2005 Today's Calculation Of Integral, 5
Calculate the following indefinite integrals.
[1] $\int (4-5\tan x)\cos x dx$
[2] $\int \frac{dx}{\sqrt[3]{(1-3x)^2}}dx$
[3] $\int x^3\sqrt{4-x^2}dx$
[4] $\int e^{-x}\sin \left(x+\frac{\pi}{4}\right)dx$
[5] $\int (3x-4)^2 dx$
2024 CMIMC Integration Bee, 9
\[\int_0^1 \frac{1-x}{x^{5/2}+x^{3/2}+x^{1/2}}\mathrm dx\]
[i]Proposed by Connor Gordon[/i]