Olympiad problems

2007 Today's Calculation Of Integral, 245

A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches 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 .$ created by kunny

2005 Today's Calculation Of Integral, 9

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

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

2010 Today's Calculation Of Integral, 616

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$. [i]2010 Hirosaki University entrance exam[/i]

2007 Today's Calculation Of Integral, 240

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

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

2007 Today's Calculation Of Integral, 230

Tags: calculus , integration , function , calculus computations , logarithm

Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.

2011 Today's Calculation Of Integral, 742

Tags: calculus , integration , calculus computations

Evaluate \[\int_0^1 \frac{1-x^2}{(1+x^2)\sqrt{1+x^4}}\ dx\]

Today's calculation of integrals, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2009 Today's Calculation Of Integral, 429

Tags: calculus , integration , trigonometry , calculus computations

Find the length of the curve expressed by the polar equation: $ r\equal{}1\plus{}\cos \theta \ (0\leq \theta \leq \pi)$.

2010 Today's Calculation Of Integral, 664

Tags: calculus , integration , trigonometry , calculus computations

For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$. Find $I_1+I_2+I_3+I_4$. [i]1992 University of Fukui entrance exam/Medicine[/i]

2010 Today's Calculation Of Integral, 575

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

For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

2007 Today's Calculation Of Integral, 234

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

For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

2010 Today's Calculation Of Integral, 584

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

Find $ \lim_{x\rightarrow \infty} \left(\int_0^x \sqrt{1\plus{}e^{2t}}\ dt\minus{}e^x\right)$.

2008 ISI B.Stat Entrance Exam, 6

Tags: limit , integration , calculus , function , real analysis , calculus computations , logarithm

Evaluate: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$

2008 Harvard-MIT Mathematics Tournament, 10

Tags: calculus , integration , calculus computations , logarithm

([b]8[/b]) Evaluate the integral $ \int_0^1\ln x \ln(1\minus{}x)\ dx$.

2010 Today's Calculation Of Integral, 601

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

Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2009 Today's Calculation Of Integral, 476

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

Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.

Today's calculation of integrals, 853

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

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

2005 Today's Calculation Of Integral, 37

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

Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

2005 Today's Calculation Of Integral, 76

Tags: calculus , integration , function , limit , inequalities , calculus computations

The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows. \[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\] Evaluate \[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]

2009 Today's Calculation Of Integral, 506

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

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$.

2009 ISI B.Stat Entrance Exam, 5

Tags: calculus , geometry , rectangle , calculus computations

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

2009 Today's Calculation Of Integral, 449

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$

2011 Today's Calculation Of Integral, 673

Tags: calculus , integration , trigonometry , calculus computations

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]

2012 Kyoto University Entry Examination, 1A

Tags: calculus , integration , symmetry , calculus computations , geometry

Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.

2010 Today's Calculation Of Integral, 598

Tags: calculus , integration , trigonometry , limit , calculus computations

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam