Olympiad problems

2010 Today's Calculation Of Integral, 610

Evaluate $\int_2^a \frac{x^a-1-xa^x\ln a}{(x^a-1)^2}dx.$ proposed by kunny

2012 Today's Calculation Of Integral, 849

Tags: calculus , integration , function , calculus computations

Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2009 Today's Calculation Of Integral, 517

Tags: calculus , integration , geometry , search , calculus computations

Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.

2009 Today's Calculation Of Integral, 404

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.

2010 Today's Calculation Of Integral, 622

Tags: calculus , integration , trigonometry , geometry , analytic geometry , function , calculus computations

For $0<k<2$, consider two curves $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$ Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$. Find the minimum value of $S(k).$ [i]2010 Nagoya Institute of Technology entrance exam[/i]

Today's calculation of integrals, 856

Tags: calculus , integration , analytic geometry , geometry , trigonometry , calculus computations

On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.

2010 Today's Calculation Of Integral, 637

Tags: calculus , integration , trigonometry , calculus computations , logarithm

For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions: (1) Calculate $I_{n+2}+I_n.$ (2) Evaluate the values of $I_1,\ I_2$ and $I_3.$ 1978 Niigata university entrance exam

2007 Today's Calculation Of Integral, 244

Tags: calculus , integration , function , geometry , calculus computations

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2012 Today's Calculation Of Integral, 821

Tags: calculus , integration , inequalities , calculus computations , logarithm

Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

2006 ISI B.Stat Entrance Exam, 1

Tags: trigonometry , analytic geometry , graphing lines , slope , calculus , calculus computations

If the normal to the curve $x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23}$ at some point makes an angle $\theta$ with the $X$-axis, show that the equation of the normal is \[y\cos\theta-x\sin\theta=a\cos 2\theta\]

2012 Today's Calculation Of Integral, 820

Tags: calculus , integration , analytic geometry , limit , calculus computations , logarithm

Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

2010 Today's Calculation Of Integral, 565

Tags: calculus , integration , function , derivative , symmetry , calculus computations

Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.

2014 Contests, 902

Tags: calculus , integration , calculus computations

For $a\geq 0$, find the minimum value of $S(a)=\int_0^1 |x^2+2ax+a^2-1|\ dx.$

2012 ISI Entrance Examination, 2

Tags: function , limit , calculus , derivative , calculus computations

Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

2007 Today's Calculation Of Integral, 252

Tags: calculus , integration , trigonometry , derivative , calculus computations

Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$

2009 Today's Calculation Of Integral, 500

Tags: calculus , integration , calculus computations

Let $ a,\ b,\ c$ be positive real numbers. Prove the following inequality. \[ \int_1^e \frac {x^{a \plus{} b \plus{} c \minus{} 1}[2(a \plus{} b \plus{} c) \plus{} (c \plus{} 2a)x^{a \minus{} b} \plus{} (a \plus{} 2b)x^{b \minus{} c} \plus{} (b \plus{} 2c)x^{c \minus{} a} \plus{}(2a \plus{} b)x^{a \minus{} c} \plus{} (2b \plus{} c)x^{b \minus{} a} \plus{} (2c \plus{} a)x^{c \minus{} b}]}{(x^a \plus{} x^b)(x^b \plus{} x^c)(x^c \plus{} x^a)}\geq a \plus{} b \plus{} c.\] I have just posted 500 th post. [color=blue]Thank you for your cooperations, mathLinkers and AOPS users.[/color] I will keep posting afterwards. Japanese Communities Modeartor kunny

2010 Today's Calculation Of Integral, 568

Tags: calculus , integration , probability , limit , calculus computations , logarithm

Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes. Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$

2009 Today's Calculation Of Integral, 409

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

2009 Today's Calculation Of Integral, 403

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.

2009 Today's Calculation Of Integral, 457

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2009 Today's Calculation Of Integral, 442

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

2005 Today's Calculation Of Integral, 54

Tags: calculus , integration , trigonometry , algebra , partial fractions , calculus computations

evaluate \[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]

Today's calculation of integrals, 885

Tags: calculus , integration , trigonometry , calculus computations , indefinite integral , logarithm

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2009 Today's Calculation Of Integral, 515

Tags: calculus , integration , trigonometry , derivative , calculus computations

Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$. Note that you are not allowed to solve in using partial differentiation here.

2011 Today's Calculation Of Integral, 732

Tags: calculus , integration , parameterization , trigonometry , calculus computations

Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.