Olympiad problems

2009 Today's Calculation Of Integral, 495

Evaluate the following definite integrals. (1) $ \int_0^{\frac {1}{2}} \frac {x^2}{\sqrt {1 \minus{} x^2}}\ dx$ (2) $ \int_0^1 \frac {1 \minus{} x}{(1 \plus{} x^2)^2}\ dx$ (3) $ \int_{ \minus{} 1}^7 \frac {dx}{1 \plus{} \sqrt [3]{1 \plus{} x}}$

2013 District Olympiad, 1

Tags: limit , integration , calculus , calculus computations

Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$

2007 Moldova National Olympiad, 12.7

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

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

Today's calculation of integrals, 875

Tags: calculus , integration , trigonometry , ratio , calculus computations

Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$

2007 Today's Calculation Of Integral, 180

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

Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2009 Today's Calculation Of Integral, 444

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

Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

2010 Today's Calculation Of Integral, 563

Tags: calculus , integration , quadratic , function , calculus computations

Determine the pair of constant numbers $ a,\ b,\ c$ such that for a quadratic function $ f(x) \equal{} x^2 \plus{} ax \plus{} b$, the following equation is identity with respect to $ x$. \[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\] .

2011 Today's Calculation Of Integral, 733

Tags: calculus , integration , limit , calculus computations

Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

2013 Today's Calculation Of Integral, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

Today's calculation of integrals, 850

Tags: calculus , integration , trigonometry , calculus computations

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

2007 Today's Calculation Of Integral, 252

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

Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$

2003 Moldova National Olympiad, 12.1

Tags: limit , calculus , calculus computations , logarithm

For every natural number $n$ let: $a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find: \[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].

2007 Today's Calculation Of Integral, 204

Tags: calculus , integration , trigonometry , special factorizations , calculus computations

Evaluate \[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]

2007 Today's Calculation Of Integral, 203

Tags: calculus , integration , trigonometry , calculus computations

Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]

2012 Today's Calculation Of Integral, 821

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

Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$

2005 Today's Calculation Of Integral, 49

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

For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$

2009 Today's Calculation Of Integral, 409

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \sqrt{\frac{x\plus{}\sqrt{x^2\plus{}1}}{x^2\plus{}1}}\ dx$.

2012 Today's Calculation Of Integral, 828

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

Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$

2009 Today's Calculation Of Integral, 404

Tags: calculus , integration , trigonometry , calculus computations

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

Today's calculation of integrals, 897

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

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.

2009 Today's Calculation Of Integral, 448

Tags: calculus , integration , calculus computations , logarithm

Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$

2005 Today's Calculation Of Integral, 82

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

Let $0<a<b$.Prove the following inequaliy. \[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]

2008 Harvard-MIT Mathematics Tournament, 2

Tags: analytic geometry , graphing lines , slope , calculus , derivative , calculus computations

([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$.

2007 Today's Calculation Of Integral, 227

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

Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$

2011 Today's Calculation Of Integral, 703

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

Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.