Olympiad problems

2005 Today's Calculation Of Integral, 25

Let $|a|<\frac{\pi}{2}$. Evaluate \[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]

Today's calculation of integrals, 882

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

Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2011 Today's Calculation Of Integral, 721

Tags: integration , function , calculus computations , calculus

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

Today's calculation of integrals, 869

Tags: calculus , integration , trigonometry , calculus computations

Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$ Answer the questions below. (1) Find $I_n.$ (2) Find $\sum_{n=1}^{\infty} I_n.$

2007 Today's Calculation Of Integral, 190

Tags: integration , calculus computations , calculus

In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis. Note you may not use double integral.

2010 Today's Calculation Of Integral, 631

Tags: calculus , logarithm , integration , calculus computations

Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

2011 Today's Calculation Of Integral, 745

Tags: calculus , integration , calculus computations , trigonometry

When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$

2005 Today's Calculation Of Integral, 45

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

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

2009 Today's Calculation Of Integral, 396

Tags: calculus , calculus computations , integration

Evaluate $ \int_0^{2008} \left(3x^2 \minus{} 8028x \plus{} 2007^2 \plus{} \frac {1}{2008}\right)\ dx$.

2010 Today's Calculation Of Integral, 618

Tags: calculus , calculus computations , integration , trigonometry

Find the minimu value of $\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.$ [i]2010 Ibaraki University entrance exam/Science[/i]

2009 Today's Calculation Of Integral, 446

Tags: calculus computations , calculus , logarithm , integration

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

2012 Today's Calculation Of Integral, 859

Tags: calculus , derivative , integration , geometry , 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 $e^3>20.$

2009 Today's Calculation Of Integral, 414

Tags: calculus , integration , calculus computations , trigonometry

Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.

2005 Today's Calculation Of Integral, 31

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

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

2009 Today's Calculation Of Integral, 465

Tags: calculus , integration , calculus computations

Compute $ \int_0^1 x^{2n\plus{}1}e^{\minus{}x^2}dx\ (n\equal{}1,\ 2,\ \cdots)$ , then use this result, prove that $ \sum_{n\equal{}0}^{\infty} \frac{1}{n!}\equal{}e$.

2005 Today's Calculation Of Integral, 85

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

Evaluate \[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\] where $ [x] $ is the integer equal to $ x $ or less than $ x $.

2007 Today's Calculation Of Integral, 199

Tags: calculus , calculus computations , integration

Let $m,\ n$ be non negative integers. Calculate \[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \] where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2012 Today's Calculation Of Integral, 774

Tags: logarithm , calculus , calculus computations , integration

Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$

2010 Today's Calculation Of Integral, 633

Tags: integration , calculus , calculus computations

Let $f(x)$ be a differentiable function. Find the value of $x$ for which \[\{f(x)\}^2+(e+1)f(x)+1+e^2-2\int_0^x f(t)dt-2f(x)\int_0^x f(t)dt+2\left\{\int_0^x f(t)dt\right\}^2\] is minimized. [i]1978 Tokyo Medical College entrance exam[/i]

2011 Today's Calculation Of Integral, 694

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

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

2011 Today's Calculation Of Integral, 709

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

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

2013 Today's Calculation Of Integral, 896

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

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

2010 Today's Calculation Of Integral, 589

Tags: calculus computations , calculus , integration

Evaluate $ \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

2010 Today's Calculation Of Integral, 622

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

For $0<k<2$, consider two curves $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$ Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$. Find the minimum value of $S(k).$ [i]2010 Nagoya Institute of Technology entrance exam[/i]

2011 Today's Calculation Of Integral, 725

Tags: calculus computations , calculus , logarithm , integration

For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$