Olympiad problems

2009 Today's Calculation Of Integral, 436

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

2009 Today's Calculation Of Integral, 487

Tags: calculus , integration , function , calculus computations

Suppose two functions $ f(x)\equal{}x^4\minus{}x,\ g(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ satisfy $ f(1)\equal{}g(1),\ f(\minus{}1)\equal{}g(\minus{}1)$. Find the values of $ a,\ b,\ c,\ d$ such that $ \int_{\minus{}1}^1 (f(x)\minus{}g(x))^2dx$ is minimal.

2008 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , derivative , trigonometry , calculus computations

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

2007 Today's Calculation Of Integral, 215

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

For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$. Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.

Today's calculation of integrals, 859

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

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

2005 Today's Calculation Of Integral, 66

Tags: calculus , integration , trigonometry , calculus computations

Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$

2005 Today's Calculation Of Integral, 59

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]

2010 Today's Calculation Of Integral, 550

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

2011 Today's Calculation Of Integral, 710

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

Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2013 Today's Calculation Of Integral, 871

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

Define sequences $\{a_n\},\ \{b_n\}$ by \[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\] (1) Find $b_n$. (2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$ (3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$

2013 Hitotsubashi University Entrance Examination, 3

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

Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2005 Today's Calculation Of Integral, 20

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

Calculate the following indefinite integrals. [1] $\int \ln (x^2-1)dx$ [2] $\int \frac{1}{e^x+1}dx$ [3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$ [4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$ [5] $\int \sqrt{1-\sin x}dx$

2013 Today's Calculation Of Integral, 895

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

In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

2007 Today's Calculation Of Integral, 179

Tags: calculus , integration , calculus computations , logarithm

Evaluate the following integrals. (1) Meiji University $\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$ (2) Tokyo University of Science $\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$

2012 Today's Calculation Of Integral, 805

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

Prove the following inequalities: (1) For $0\leq x\leq 1$, \[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\] (2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$

2005 Today's Calculation Of Integral, 46

Tags: calculus , integration , calculus computations , logarithm

Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$

2009 Today's Calculation Of Integral, 511

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

Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

2005 Today's Calculation Of Integral, 29

Tags: calculus , integration , trigonometry , calculus computations

Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

2010 Today's Calculation Of Integral, 635

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

Suppose that a function $f(x)$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii). (i) $f'(x)$ is continuous. (ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$. (iii) $f(0)=-1$ Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ holds for $-\frac{\pi}{2}<\theta <\frac{\pi}{2}$, then find $f(x)$. [i]1975 Waseda University entrance exam/Science and Technology[/i]

2005 Today's Calculation Of Integral, 26

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

2010 Today's Calculation Of Integral, 642

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\frac{\pi}{6}} \frac{(\tan ^ 2 2x)\sqrt{\cos 2x}+2}{(\cos ^ 2 x)\sqrt{\cos 2x}}dx.\] Own

2011 Today's Calculation Of Integral, 766

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

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2005 Today's Calculation Of Integral, 73

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

Find the minimum value of $\int_0^{\pi} (a\sin x+b\sin 2x+c\sin 3x-x)^2\ dx$

2009 Today's Calculation Of Integral, 477

Tags: calculus , integration , calculus computations

Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$. Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.

2011 Today's Calculation Of Integral, 694

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

Prove the following inequality: \[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\] created by kunny