Found problems: 1687
2010 Today's Calculation Of Integral, 554
Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.
2005 Today's Calculation Of Integral, 29
Let $a$ be a real number.
Evaluate
\[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]
1983 Miklós Schweitzer, 5
Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or
decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists.
[i]T. Krisztin[/i]
Today's calculation of integrals, 879
Evaluate the integrals as follows.
(1) $\int \frac{x^2}{2-x}\ dx$
(2) $\int \sqrt[3]{x^5+x^3}\ dx$
(3) $\int_0^1 (1-x)\cos \pi x\ dx$
2012 Today's Calculation Of Integral, 807
Define a sequence $a_n$ satisfying :
\[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\]
Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$
2007 Today's Calculation Of Integral, 192
Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$
2007 ISI B.Stat Entrance Exam, 3
Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$,
\[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]
2011 Today's Calculation Of Integral, 689
Let $C: y=x^2+ax+b$ be a parabola passing through the point $(1,\ -1)$. Find the minimum volume of the figure enclosed by $C$ and the $x$ axis by a rotation about the $x$ axis.
Proposed by kunny
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$
2007 Today's Calculation Of Integral, 219
Let $ f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0)$.
Find $ \lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}$.
2022 JHMT HS, 9
There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and
\[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \]
for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.
1949 Miklós Schweitzer, 7
Find the complex numbers $ z$ for which the series
\[ 1 \plus{} \frac {z}{2!} \plus{} \frac {z(z \plus{} 1)}{3!} \plus{} \frac {z(z \plus{} 1)(z \plus{} 2)}{4!} \plus{} \cdots \plus{} \frac {z(z \plus{} 1)\cdots(z \plus{} n)}{(n \plus{} 2)!} \plus{} \cdots\]
converges and find its sum.
2008 Harvard-MIT Mathematics Tournament, 2
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?
2017 BMT Spring, 8
The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$.
(Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)
2011 Today's Calculation Of Integral, 706
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad \\
z\leq y^2 &\quad
\end{array}
\right.\]
Find the volume of the solid $V$.
Today's calculation of integrals, 860
For a function $f(x)\ (x\geq 1)$ satisfying $f(x)=(\log_e x)^2-\int_1^e \frac{f(t)}{t}dt$, answer the questions as below.
(a) Find $f(x)$ and the $y$-coordinate of the inflection point of the curve $y=f(x)$.
(b) Find the area of the figure bounded by the tangent line of $y=f(x)$ at the point $(e,\ f(e))$, the curve $y=f(x)$ and the line $x=1$.
2011 Putnam, B1
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
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 $0\leq a <2\pi$
2005 Today's Calculation Of Integral, 59
Evaluate
\[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]
PEN H Problems, 44
For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.
2013 India National Olympiad, 3
Let $a,b,c,d \in \mathbb{N}$ such that $a \ge b \ge c \ge d $. Show that the equation $x^4 - ax^3 - bx^2 - cx -d = 0$ has no integer solution.
2007 AMC 12/AHSME, 19
Triangles $ ABC$ and $ ADE$ have areas $ 2007$ and $ 7002,$ respectively, with $ B \equal{} (0,0),$ $ C \equal{} (223,0),$ $ D \equal{} (680,380),$ and $ E \equal{} (689,389).$ What is the sum of all possible x-coordinates of $ A?$
$ \textbf{(A)}\ 282 \qquad \textbf{(B)}\ 300 \qquad \textbf{(C)}\ 600 \qquad \textbf{(D)}\ 900 \qquad \textbf{(E)}\ 1200$
2023 CMIMC Integration Bee, 2
\[\int_0^1 \frac{1}{x+\sqrt x}\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2010 Today's Calculation Of Integral, 551
In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.
1980 USAMO, 3
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.