Olympiad problems

2007 Today's Calculation Of Integral, 246

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

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

2005 Today's Calculation Of Integral, 17

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

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

2005 Today's Calculation Of Integral, 87

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

Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized. \[ \int_0^{\pi} |\sin x-ax|\ dx \]

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] $\int \sqrt{1-3x}\ dx$

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.$

2012 Today's Calculation Of Integral, 844

Tags: calculus , integration , limit , calculus computations

Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$ Find $\lim_{n\to\infty} n^2I_n.$

2007 Today's Calculation Of Integral, 245

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

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

2007 Today's Calculation Of Integral, 228

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

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

2010 Today's Calculation Of Integral, 632

Tags: calculus , integration , limit , trigonometry , floor function , ceiling function , calculus computations

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

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, 80

Tags: calculus , integration , algebra , function , domain , ratio , calculus computations

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

2010 Today's Calculation Of Integral, 665

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

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]

2011 Today's Calculation Of Integral, 700

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

2013 Today's Calculation Of Integral, 876

Tags: calculus , integration , calculus computations , definite integral , function

Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition : 1) $f(-1)\geq f(1).$ 2) $x+f(x)$ is non decreasing function. 3) $\int_{-1}^ 1 f(x)\ dx=0.$ Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

2012 Today's Calculation Of Integral, 831

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

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 $\lim_{x\to\infty} I_n(x)$.

2012 Hitotsubashi University Entrance Examination, 2

Tags: function , calculus , calculus computations

Let $a\geq 0$ be constant. Find the number of Intersection points of the graph of the function $y=x^3-3a^2x$ and the figure expressed by the equation $|x|+|y|=2$.

2007 Today's Calculation Of Integral, 195

Tags: calculus , integration , function , real analysis , algebra , partial fractions , calculus computations

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

2007 Today's Calculation Of Integral, 173

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

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

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]

2011 Today's Calculation Of Integral, 721

Tags: calculus , integration , function , calculus computations

For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.

2005 Today's Calculation Of Integral, 6

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

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

2009 Today's Calculation Of Integral, 475

Tags: calculus , integration , calculus computations

For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$

2005 Today's Calculation Of Integral, 27

Tags: calculus , integration , trigonometry , quadratic , algebra , calculus computations

Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

Today's calculation of integrals, 893

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

Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2009 Today's Calculation Of Integral, 467

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

Let the curve $ C: y\equal{}x\sqrt{9\minus{}x^2}\ (x\geq 0)$. (1) Find the maximum value of $ y$. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$ axis. (3) Find the volume of the solid generated by rotation of the figure about the $ y$ axis.