Found problems: 1687
2022 CMIMC Integration Bee, 4
\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2011 Today's Calculation Of Integral, 673
Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$
[i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]
2009 Today's Calculation Of Integral, 485
In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis.
Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.
2005 All-Russian Olympiad, 2
Find the number of subsets $A\subset M=\{2^0,\,2^1,\,2^2,\dots,2^{2005}\}$ such that equation $x^2-S(A)x+S(B)=0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B=M\setminus A$ ($A$ and $B$ are not empty).
2012 Today's Calculation Of Integral, 853
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
2011 Today's Calculation Of Integral, 741
Evaluate
\[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]
2012 Today's Calculation Of Integral, 818
For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$
2013 Today's Calculation Of Integral, 876
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
2007 Princeton University Math Competition, 5
Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $.
PEN E Problems, 27
Prove that for each positive integer $n$, there exist $n$ consecutive positive integers none of which is an integral power of a prime number.
2020 Jozsef Wildt International Math Competition, W60
Compute
$$\int\frac{(\sin x+\cos x)(4-2\sin2x-\sin^22x)e^x}{\sin^32x}dx$$
where $x\in\left(0,\frac\pi2\right)$.
[i]Proposed by Mihály Bencze[/i]
1975 Canada National Olympiad, 4
For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
2012 Today's Calculation Of Integral, 801
Answer the following questions:
(1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$.
Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$.
(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$
2007 Today's Calculation Of Integral, 208
Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.
2011 Today's Calculation Of Integral, 733
Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2
For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function.
Prove that :
(i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$
(ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$
2007 Today's Calculation Of Integral, 239
Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$
2006 Taiwan National Olympiad, 3
If positive integers $p,q,r$ are such that the quadratic equation $px^2-qx+r=0$ has two distinct real roots in the open interval $(0,1)$, find the minimum value of $p$.
2009 Today's Calculation Of Integral, 481
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$.
Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$
2011 BMO TST, 2
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
2006 Turkey Team Selection Test, 2
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?
2010 Romania Team Selection Test, 1
A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$.
[i]AMM Magazine[/i]
2010 Today's Calculation Of Integral, 661
Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$.
(1) Find the 36th term.
(2) Find $\int x^2\ln x\ dx$.
(3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part?
If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$.
[i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]
2011 Today's Calculation Of Integral, 701
Evaluate
\[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]
2011 Today's Calculation Of Integral, 692
Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$.
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