Olympiad problems

2007 Today's Calculation Of Integral, 231

Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.

2005 Today's Calculation Of Integral, 65

Tags: calculus , integration , calculus computations

Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

2009 Today's Calculation Of Integral, 518

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

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

2011 Today's Calculation Of Integral, 748

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

Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

2013 District Olympiad, 4

Tags: function , limit , calculus , calculus computations

Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function. a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$. b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.

2009 Today's Calculation Of Integral, 452

Tags: calculus , integration , calculus computations , logarithm

Let $ a,\ b$ are postive constant numbers. (1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$ (2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$

2010 Today's Calculation Of Integral, 600

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

2010 Today's Calculation Of Integral, 656

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

Find $\lim_{n\to\infty} n\int_0^{\frac{\pi}{2}} \frac{1}{(1+\cos x)^n}dx\ (n=1,\ 2,\ \cdots).$

2012 Today's Calculation Of Integral, 827

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

Find $\lim_{n\to\infty}\sum_{k=0}^{\infty} \int_{2k\pi}^{(2k+1)\pi} xe^{-x}\sin x\ dx.$

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 $m.$

2007 Today's Calculation Of Integral, 178

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

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

2011 Today's Calculation Of Integral, 751

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

Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$

Today's calculation of integrals, 765

Tags: calculus , integration , function , limit , calculus computations

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.