Olympiad problems

2010 Today's Calculation Of Integral, 573

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

Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

2012 Today's Calculation Of Integral, 852

Tags: calculus computations , calculus , absolute value , polynomial , integration , algebra , trigonometry

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, 610

Tags: logarithm , calculus computations , calculus , integration

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

2007 Today's Calculation Of Integral, 174

Tags: geometry , parameterization , calculus computations , calculus , integration

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

2005 Today's Calculation Of Integral, 71

Tags: calculus computations , integration , calculus

Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

2009 Today's Calculation Of Integral, 411

Tags: geometry , calculus , integration , calculus computations

Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

2010 Today's Calculation Of Integral, 609

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

Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number. 1972 Tokyo University of Education entrance exam

2005 Today's Calculation Of Integral, 33

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

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

2012 Today's Calculation Of Integral, 823

Tags: geometry , rotation , trigonometry , calculus computations , calculus , integration , geometric transformation

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

2011 Today's Calculation Of Integral, 732

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

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

2010 Today's Calculation Of Integral, 665

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

Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$. [i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]

2012 Today's Calculation Of Integral, 812

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

Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$

2012 Today's Calculation Of Integral, 835

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

Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

2005 Today's Calculation Of Integral, 10

Tags: calculus computations , calculus , trigonometry , integration

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

2011 Today's Calculation Of Integral, 690

Tags: calculus computations , calculus , integration , vector , trigonometry

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

2009 Today's Calculation Of Integral, 469

Tags: calculus , calculus computations , trigonometry , integration

Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.

2005 Today's Calculation Of Integral, 5

Tags: integration , trigonometry , calculus , calculus computations

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

2012 Today's Calculation Of Integral, 831

Tags: calculus computations , limit , integration , calculus , induction

Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

2010 Today's Calculation Of Integral, 585

Tags: calculus computations , logarithm , integration , calculus

Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

2008 Harvard-MIT Mathematics Tournament, 3

Tags: logarithm , calculus , integration , calculus computations

([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.

2005 Today's Calculation Of Integral, 88

Tags: calculus computations , calculus , function , integration

A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$ Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$

2009 Today's Calculation Of Integral, 441

Tags: logarithm , integration , calculus , calculus computations

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

2007 Today's Calculation Of Integral, 206

Tags: calculus , integration , calculus computations

Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$

2010 Today's Calculation Of Integral, 564

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

In the coordinate plane with $ O(0,\ 0)$, consider the function $ C: \ y \equal{} \frac 12x \plus{} \sqrt {\frac 14x^2 \plus{} 2}$ and two distinct points $ P_1(x_1,\ y_1),\ P_2(x_2,\ y_2)$ on $ C$. (1) Let $ H_i\ (i \equal{} 1,\ 2)$ be the intersection points of the line passing through $ P_i\ (i \equal{} 1,\ 2)$, parallel to $ x$ axis and the line $ y \equal{} x$. Show that the area of $ \triangle{OP_1H_1}$ and $ \triangle{OP_2H_2}$ are equal. (2) Let $ x_1 < x_2$. Express the area of the figure bounded by the part of $ x_1\leq x\leq x_2$ for $ C$ and line segments $ P_1O,\ P_2O$ in terms of $ y_1,\ y_2$.

2009 Today's Calculation Of Integral, 504

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

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.