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, 681
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

2009 Today's Calculation Of Integral, 506
Tags: calculus , integration , function , parabola , geometry , calculus computations , conic
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

2005 Today's Calculation Of Integral, 70
Tags: calculus , integration , trigonometry , function , calculus computations
Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$

1962 Vietnam National Olympiad, 2
Tags: calculus , derivative , function , calculus computations
Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.

2010 Today's Calculation Of Integral, 567
Tags: calculus , integration , analytic geometry , geometry , trigonometry , calculus computations
Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.

2012 Kyoto University Entry Examination, 1A
Tags: calculus , integration , geometry , symmetry , calculus computations
Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

2010 Today's Calculation Of Integral, 527
Tags: calculus , integration , calculus computations
Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

2007 Moldova National Olympiad, 11.2
Tags: limit , calculus , calculus computations , logarithm
Define $a_{n}$ as satisfying: $\left(1+\frac{1}{n}\right)^{n+a_{n}}=e$. Find $\lim_{n\rightarrow\infty}a_{n}$.

2005 Today's Calculation Of Integral, 40
Tags: calculus , integration , calculus computations
Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

2005 Today's Calculation Of Integral, 8
Tags: calculus , integration , trigonometry , function , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int x(x^2+3)^2 dx$ [2] $\int \ln (x+2) dx$ [3] $\int x\cos x dx$ [4] $\int \frac{dx}{(x+2)^2}dx$ [5] $\int \frac{x-1}{x^2-2x+3}dx$

2009 Today's Calculation Of Integral, 518
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

2012 Today's Calculation Of Integral, 855
Tags: calculus , integration , function , limit , calculus computations
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2010 Today's Calculation Of Integral, 639
Tags: calculus , integration , calculus computations
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

2009 Today's Calculation Of Integral, 428
Tags: calculus , integration , algebra , polynomial , calculus computations
Let $ f(x)$ be a polynomial and $ C$ be a real number. Find the $ f(x)$ and $ C$ such that $ \int_0^x f(y)dy\plus{}\int_0^1 (x\plus{}y)^2f(y)dy\equal{}x^2\plus{}C$.

2007 Today's Calculation Of Integral, 243
Tags: calculus , integration , function , geometry , calculus computations
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.

2009 Today's Calculation Of Integral, 400
Tags: calculus , integration , function , calculus computations , logarithm
(1) A function is defined $ f(x) \equal{} \ln (x \plus{} \sqrt {1 \plus{} x^2})$ for $ x\geq 0$. Find $ f'(x)$. (2) Find the arc length of the part $ 0\leq \theta \leq \pi$ for the curve defined by the polar equation: $ r \equal{} \theta\ (\theta \geq 0)$. Remark: [color=blue]You may not directly use the integral formula of[/color] $ \frac {1}{\sqrt {1 \plus{} x^2}},\ \sqrt{1 \plus{} x^2}$ here.

2007 Today's Calculation Of Integral, 201
Tags: calculus , integration , limit , function , calculus computations
Evaluate the following definite integral. \[\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx\]

2011 Today's Calculation Of Integral, 736
Tags: calculus , integration , derivative , calculus computations , logarithm
Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

2007 Today's Calculation Of Integral, 220
Tags: calculus , integration , inequalities , calculus computations
Prove that $ \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx$.

2010 Today's Calculation Of Integral, 543
Tags: calculus , integration , function , trigonometry , calculus computations
Let $ y$ be the function of $ x$ satisfying the differential equation $ y'' \minus{} y \equal{} 2\sin x$. (1) Let $ y \equal{} e^xu \minus{} \sin x$, find the differential equation with which the function $ u$ with respect to $ x$ satisfies. (2) If $ y(0) \equal{} 3,\ y'(0) \equal{} 0$, then determine $ y$.

2010 Today's Calculation Of Integral, 544
Tags: calculus , integration , function , calculus computations
(1) Evaluate $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}}( x^2\minus{}1)dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)^2dx,\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)^2dx$. (2) If a linear function $ f(x)$ satifies $ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\minus{}1)f(x)dx\equal{}5\sqrt{3},\ \int_{\minus{}\sqrt{3}}^{\sqrt{3}} (x\plus{}1)f(x)dx\equal{}3\sqrt{3}$, then we have $ f(x)\equal{}\boxed{\ A\ }(x\minus{}1)\plus{}\boxed{\ B\ }(x\plus{}1)$, thus we have $ f(x)\equal{}\boxed{\ C\ }$.

2013 Today's Calculation Of Integral, 881
Tags: calculus , integration , trigonometry , calculus computations , definite integral
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2005 Today's Calculation Of Integral, 45
Tags: calculus , integration , function , trigonometry , calculus computations
Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

2005 Today's Calculation Of Integral, 59
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

Today's calculation of integrals, 899
Tags: calculus , integration , limit , calculus computations
Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]