This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 713

2011 Today's Calculation Of Integral, 704
Tags: calculus , integration , function , induction , calculus computations
A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

2009 Today's Calculation Of Integral, 484
Tags: calculus , integration , geometry , limit , trapezoid , calculus computations , logarithm
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$. (1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively. (2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.

2003 Moldova National Olympiad, 12.8
Tags: limit , function , calculus , calculus computations
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by $F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]

2007 Today's Calculation Of Integral, 202
Tags: calculus , integration , trigonometry , quadratic , inequalities , calculus computations
Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]

2011 Today's Calculation Of Integral, 731
Tags: calculus , integration , parabola , geometry , calculus computations , conic
Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.

2009 Today's Calculation Of Integral, 499
Tags: calculus , integration , trigonometry , symmetry , function , calculus computations
Evaluate \[ \int_0^{\pi} (\sqrt[2009]{\cos x}\plus{}\sqrt[2009]{\sin x}\plus{}\sqrt[2009]{\tan x})\ dx.\]

2012 Today's Calculation Of Integral, 794
Tags: calculus , integration , function , trigonometry , calculus computations
Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

2005 Today's Calculation Of Integral, 68
Tags: calculus , integration , derivative , calculus computations , logarithm
Find the minimum value of $\int_1^e \left|\ln x-\frac{a}{x}\right|dx\ (0\leq a\leq e)$

2005 Today's Calculation Of Integral, 1
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

2005 Today's Calculation Of Integral, 74
Tags: calculus , integration , calculus computations , logarithm
$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

Today's calculation of integrals, 767
Tags: calculus , integration , trigonometry , calculus computations
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2007 Today's Calculation Of Integral, 190
Tags: calculus , integration , calculus computations
In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis. Note you may not use double integral.

2005 Today's Calculation Of Integral, 87
Tags: calculus , integration , trigonometry , derivative , calculus computations
Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

2012 Today's Calculation Of Integral, 830
Tags: calculus , integration , limit , function , calculus computations , logarithm
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

2013 District Olympiad, 1
Tags: limit , inequalities , calculus , calculus computations
Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

2011 ISI B.Stat Entrance Exam, 4
Tags: function , integration , inequalities , calculus , calculus computations
Let $f$ be a twice differentiable function on the open interval $(-1,1)$ such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0$ and $f''(x) \le f(x)$, for all $x\ge 0$. Show that $f'(0) \ge -\sqrt2$.

2009 Today's Calculation Of Integral, 446
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_0^1 \frac{(1\minus{}2x)e^{x}\plus{}(1\plus{}2x)e^{\minus{}x}}{(e^x\plus{}e^{\minus{}x})^3}\ dx.$

2005 Today's Calculation Of Integral, 67
Tags: calculus , integration , geometry , calculus computations
Evaluate \[\frac{2005\displaystyle \int_0^{1002}\frac{dx}{\sqrt{1002^2-x^2}+\sqrt{1003^2-x^2}}+\int_{1002}^{1003}\sqrt{1003^2-x^2}dx}{\displaystyle \int_0^1\sqrt{1-x^2}dx}\]

2011 Today's Calculation Of Integral, 744
Tags: calculus , integration , inequalities , calculus computations
Let $a,\ b$ be real numbers. If $\int_0^3 (ax-b)^2dx\leq 3$ holds, then find the values of $a,\ b$ such that $\int_0^3 (x-3)(ax-b)dx$ is minimized.

2011 Today's Calculation Of Integral, 714
Tags: calculus , integration , geometry , calculus computations
Find the area enclosed by the graph of $a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).$

2010 Today's Calculation Of Integral, 660
Tags: calculus , integration , calculus computations , logarithm
Let $a,\ b$ be given positive constants. Evaluate \[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\] Own

2009 Today's Calculation Of Integral, 413
Tags: calculus , integration , trigonometry , derivative , calculus computations
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.

Today's calculation of integrals, 888
Tags: calculus , integration , analytic geometry , trigonometry , geometry , calculus computations
In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.

2007 Today's Calculation Of Integral, 203
Tags: calculus , integration , trigonometry , calculus computations
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]

2007 Today's Calculation Of Integral, 205
Tags: calculus , integration , calculus computations , logarithm
Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]