Olympiad problems

2010 Today's Calculation Of Integral, 629

Evaluate $\int_0^{\infty} \frac{1}{e^{x}(1+e^{4x})}dx.$

2009 Today's Calculation Of Integral, 512

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

Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$

2011 Today's Calculation Of Integral, 749

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

Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.

2012 Today's Calculation Of Integral, 841

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

Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

2012 Today's Calculation Of Integral, 770

Tags: calculus , integration , calculus computations

Find the value of $a$ such that : \[101a=6539\int_{-1}^1 \frac{x^{12}+31}{1+2011^{x}}\ dx.\]

2011 Today's Calculation Of Integral, 682

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

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

2009 Today's Calculation Of Integral, 412

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

Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$. (1) Find $ I_0,\ I_{\minus{}1},\ I_2$. (2) Find $ I_1$. (3) Express $ I_{n\plus{}2}$ in terms of $ I_n$. (4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$. (5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results. You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.

2013 Today's Calculation Of Integral, 863

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

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

2010 Today's Calculation Of Integral, 532

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

For a curve $ C: y \equal{} x\sqrt {9 \minus{} x^2}\ (x\geq 0)$, (1) Find the maximum value of the function. (2) Find the area of the figure bounded by the curve $ C$ and the $ x$-axis. (3) Find the volume of the solid by revolution of the figure in (2) around the $ y$-axis. Please find the volume without using cylindrical shells for my students. Last Edited.

2007 Today's Calculation Of Integral, 196

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

Calculate \[\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \]

2011 Today's Calculation Of Integral, 690

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

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

2010 Today's Calculation Of Integral, 612

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

For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality. \[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]

2011 Today's Calculation Of Integral, 674

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.$ [i]2011 Doshisya University entrance exam/Science and Technology[/i]

2005 Today's Calculation Of Integral, 55

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

Evaluate \[\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( n=1,2,\cdots)\]

2005 Today's Calculation Of Integral, 83

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\sum_{n=1}^{\infty} \int_{2n\pi}^{2(n+1)\pi} \frac{x\sin x+\cos x}{x^2}\ dx\ (n=1,2,\cdots)\]

Today's calculation of integrals, 864

Tags: calculus , integration , inequalities , limit , calculus computations

Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2010 Today's Calculation Of Integral, 618

Tags: calculus , integration , trigonometry , calculus computations

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]

2007 Today's Calculation Of Integral, 187

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

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 Waseda University Entrance Examination, 4

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

For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

2005 Today's Calculation Of Integral, 10

Tags: calculus , integration , trigonometry , calculus computations

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

2007 Today's Calculation Of Integral, 174

Tags: calculus , integration , geometry , parameterization , calculus computations

Let $a$ be a positive number. Assume that the parameterized curve $C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)$ is touched to $x$ axis. (1) Find the value of $a.$ (2) Find the area of the part which is surrounded by two straight lines $y=0, y=x$ and the curve $C.$

2012 Today's Calculation Of Integral, 842

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

Let $S_n=\int_0^{\pi} \sin ^ n x\ dx\ (n=1,\ 2,\ ,\ \cdots).$ Find $\lim_{n\to\infty} nS_nS_{n+1}.$

2012 Kyoto University Entry Examination, 1

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

Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2010 Today's Calculation Of Integral, 663

Tags: calculus , integration , geometry , calculus computations

Given are the curve $y=x^2+x-2$ and a curve which is obtained by tranfering the curve symmetric with respect to the point $(p,\ 2p)$. Let $p$ change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves. [i]1978 Nagasaki University entrance exam/Economics[/i]

2009 Today's Calculation Of Integral, 439

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

Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$. Note that you may not directively use double integral here for Japanese high school students who don't study it.