Olympiad problems

2010 Today's Calculation Of Integral, 586

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

Evaluate $ \int_0^1 \frac{x^{14}}{x^2\plus{}1}\ dx$.

2005 Today's Calculation Of Integral, 48

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

Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

2011 Today's Calculation Of Integral, 762

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

Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

2010 Today's Calculation Of Integral, 524

Tags: calculus , integration , function , calculus computations

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2010 Today's Calculation Of Integral, 596

Tags: calculus , integration , trigonometry , calculus computations

Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$ 2009 Shimane University entrance exam/Medicine

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.

2008 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , derivative , trigonometry , calculus computations

([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).

2007 Today's Calculation Of Integral, 235

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

Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.

2007 ISI B.Stat Entrance Exam, 2

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

Use calculus to find the behaviour of the function \[y=e^x\sin{x} \ \ \ \ \ \ \ -\infty <x< +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima and points of inflection in your graph.

2009 Today's Calculation Of Integral, 520

Tags: calculus , integration , calculus computations

Let $ a,\ b,\ c$ be postive constants. Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.

2007 Today's Calculation Of Integral, 251

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

Evaluate $ \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).$

2011 Today's Calculation Of Integral, 725

Tags: calculus , integration , calculus computations , logarithm

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

2005 Today's Calculation Of Integral, 1

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

Calculate the following indefinite integral. [1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$ [2] $\int \sin x\cos 3x dx$ [3] $\int \sin 2x\sin 3x dx$ [4] $\int \frac{dx}{4x^2-12x+9}$ [5] $\int \cos ^4 x dx$

2013 Today's Calculation Of Integral, 866

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

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2010 Today's Calculation Of Integral, 650

Tags: calculus , integration , polynomial , calculus computations , algebra

Find the values of $p,\ q,\ r\ (-1<p<q<r<1)$ such that for any polynomials with degree$\leq 2$, the following equation holds: \[\int_{-1}^p f(x)\ dx-\int_p^q f(x)\ dx+\int_q^r f(x)\ dx-\int_r^1 f(x)\ dx=0.\] [i]1995 Hitotsubashi University entrance exam/Law, Economics etc.[/i]

2010 Today's Calculation Of Integral, 646

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

Evaluate \[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\] Own

2005 Today's Calculation Of Integral, 28

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\frac{\pi}{4}} \frac{x\cos 5x}{\cos x}dx\]

2007 Today's Calculation Of Integral, 244

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

A quartic funtion $ y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta \ (\alpha < \beta ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2012 Today's Calculation Of Integral, 825

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

Answer the following questions. (1) For $x\geq 0$, show that $x-\frac{x^3}{6}\leq \sin x\leq x.$ (2) For $x\geq 0$, show that $\frac{x^3}{3}-\frac{x^5}{30}\leq \int_0^x t\sin t\ dt\leq \frac{x^3}{3}.$ (3) Find the limit \[\lim_{x\rightarrow 0} \frac{\sin x-x\cos x}{x^3}.\]

2007 ISI B.Stat Entrance Exam, 7

Tags: geometry , 3d geometry , prism , trigonometry , calculus , calculus computations

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

2011 Today's Calculation Of Integral, 685