Olympiad problems

2009 Today's Calculation Of Integral, 421

Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions. (1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$. (2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal. (3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.

2009 Today's Calculation Of Integral, 404

Tags: calculus computations , trigonometry , integration , calculus

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

2005 Today's Calculation Of Integral, 74

Tags: logarithm , calculus , calculus computations , integration

$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$. Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value. \[\int_a^b (px+q-\ln x)dx\]

2011 Today's Calculation Of Integral, 736

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

Evaluate \[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]

2010 Today's Calculation Of Integral, 534

Tags: calculus , function , integration , calculus computations

Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.

2011 Today's Calculation Of Integral, 681

Tags: calculus computations , calculus , trigonometry , integration

Evaluate $\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.$ [i]2011 University of Occupational and Environmental Health/Medicine　entrance exam[/i]

2005 Today's Calculation Of Integral, 73

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

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

2012 Today's Calculation Of Integral, 814

Tags: calculus , calculus computations , integration , geometry

Find the area of the region bounded by $C: y=-x^4+8x^3-18x^2+11$ and the tangent line which touches $C$ at distinct two points.

2011 Today's Calculation Of Integral, 726

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

Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.

2007 Today's Calculation Of Integral, 237

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

Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

2005 Today's Calculation Of Integral, 3

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

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

2012 Today's Calculation Of Integral, 794

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

Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$. Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.

2010 Today's Calculation Of Integral, 554

Tags: calculus , integration , logarithm , calculus computations

Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.

2005 Today's Calculation Of Integral, 29

Tags: calculus , calculus computations , trigonometry , integration

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

Today's calculation of integrals, 879

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

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

2012 Today's Calculation Of Integral, 807

Tags: integration , limit , calculus computations , calculus

Define a sequence $a_n$ satisfying : \[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\] Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$

2007 Today's Calculation Of Integral, 192

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

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

2007 ISI B.Stat Entrance Exam, 3

Tags: calculus , calculus computations , function , integration

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$, \[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]

2011 Today's Calculation Of Integral, 706

Tags: integration , calculus computations , calculus

In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

2005 Today's Calculation Of Integral, 70

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

Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$

2005 Today's Calculation Of Integral, 59

Tags: trigonometry , calculus , calculus computations , integration

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

2010 Today's Calculation Of Integral, 551

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

In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$.

2007 Moldova National Olympiad, 12.7

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

Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]

2007 Today's Calculation Of Integral, 187

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

For a constant $a,$ let $f(x)=ax\sin x+x+\frac{\pi}{2}.$ Find the range of $a$ such that $\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).$

2012 Today's Calculation Of Integral, 799

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

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points