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, 734
Tags: calculus , integration , calculus computations , logarithm
Find the extremum of $f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)$.

2011 Today's Calculation Of Integral, 701
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]

2012 Today's Calculation Of Integral, 815
Tags: calculus , integration , trigonometry , function , calculus computations
Prove that : $\left|\sum_{i=0}^n \left(1-\pi \sin \frac{i\pi}{4n}\cos \frac{i\pi}{4n}\right)\right|<1.$

2005 Today's Calculation Of Integral, 42
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

2005 Today's Calculation Of Integral, 60
Tags: calculus , integration , trigonometry , calculus computations
Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$ Evaluate \[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]

2009 Today's Calculation Of Integral, 510
Tags: calculus , integration , trigonometry , function , derivative , calculus computations
(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

2010 Today's Calculation Of Integral, 640
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{\frac{\pi}{4}} \frac{1}{1-\sin x}\sqrt{\frac{\cos x}{1+\cos x+\sin x}}dx.$ Own

2009 Today's Calculation Of Integral, 488
Tags: calculus , integration , trigonometry , inequalities , calculus computations , logarithm
For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$

2011 Today's Calculation Of Integral, 713
Tags: calculus , integration , limit , calculus computations
If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$

2012 Today's Calculation Of Integral, 845
Tags: calculus , integration , calculus computations
Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$ Find the minimum value of $I(t)\ (t>1).$

2013 Today's Calculation Of Integral, 884
Tags: calculus , integration , trigonometry , inequalities , calculus computations , integral inequality
Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

2011 ISI B.Stat Entrance Exam, 8
Tags: integration , trigonometry , calculus , calculus computations
Let \[I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} \, dx , \ \ \ \ n=1,2,3,4\] Arrange $I_1, I_2, I_3, I_4$ in increasing order of magnitude. Justify your answer.

2012 Today's Calculation Of Integral, 852
Tags: calculus , integration , trigonometry , algebra , polynomial , absolute value , calculus computations
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows. (1) $g_n(x)=(1+x)^n$ (2) $g_n(x)=\sin n\pi x$ (3) $g_n(x)=e^{nx}$

2010 Today's Calculation Of Integral, 545
Tags: calculus , integration , function , calculus computations
(1) Evaluate $ \int_0^1 xe^{x^2}dx$. (2) Let $ I_n\equal{}\int_0^1 x^{2n\minus{}1}e^{x^2}dx$. Express $ I_{n\plus{}1}$ in terms of $ I_n$.

2005 Today's Calculation Of Integral, 83
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

2009 Today's Calculation Of Integral, 507
Tags: calculus , integration , calculus computations , logarithm
Evaluate \[ \int_e^{e^{2009}} \frac{1}{x}\left\{1\plus{}\frac{1\minus{}\ln x}{\ln x\cdot \ln \frac{x}{\ln (\ln x)}}\right\}\ dx\]

2010 Today's Calculation Of Integral, 635
Tags: calculus , integration , function , trigonometry , derivative , calculus computations
Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii). (i) $f'(x)$ is continuous. (ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$. (iii) $f(0)=-1$ Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$. [i]1975 Waseda University entrance exam/Science and Technology[/i]

2010 Today's Calculation Of Integral, 542
Tags: calculus , integration , function , calculus computations
Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.

2007 Today's Calculation Of Integral, 214
Tags: calculus , integration , geometry , calculus computations
Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.

2005 Today's Calculation Of Integral, 20
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

2013 Today's Calculation Of Integral, 897
Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2011 Today's Calculation Of Integral, 766
Tags: calculus , integration , function , trigonometry , inequalities , calculus computations
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2009 Today's Calculation Of Integral, 474
Tags: calculus , integration , trigonometry , calculus computations
Calculate the following indefinite integrals. (1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$ (2) $ \int \sin 2x\cos 2x\cos 4x\ dx$ (3) $ \int xe^{x}dx$ (4) $ \int 5^{x}dx$

2009 Today's Calculation Of Integral, 422
Tags: calculus , integration , probability , calculus computations
There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time. Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.

2010 Today's Calculation Of Integral, 613
Tags: calculus , integration , geometry , calculus computations , logarithm
Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]