Olympiad problems

2007 Today's Calculation Of Integral, 202

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

Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]

2007 Today's Calculation Of Integral, 208

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

Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.

2012 Today's Calculation Of Integral, 811

Tags: calculus , integration , trigonometry , calculus computations

Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$

2010 Today's Calculation Of Integral, 598

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

For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

2010 Today's Calculation Of Integral, 627

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

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$ [i]Proposed by kunny[/i]

2011 Today's Calculation Of Integral, 735

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

Evaluate the following definite integrals: (a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$ (b) $\int_0^{\frac 13} xe^{3x}\ dx$ (c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$ (d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2011 Today's Calculation Of Integral, 679

Tags: calculus , integration , calculus computations

Find $\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}$. [i]2011 Hosei University entrance exam/Design and Enginerring[/i]

2011 Today's Calculation Of Integral, 704

Tags: calculus , integration , function , induction , calculus computations

A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

2010 Today's Calculation Of Integral, 628

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

(1) Evaluate the following definite integrals. (a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$ (b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$ (c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$ (2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that \[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\] , then find the volume of the solid. [i]1984 Yamanashi Medical University entrance exam[/i] Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them. Thanks in advance. kunny

2009 Today's Calculation Of Integral, 445

Tags: calculus , integration , calculus computations , logarithm

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.$

2014 Contests, 901

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

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

2010 Today's Calculation Of Integral, 641

Tags: calculus , integration , calculus computations , logarithm

Evaluate \[\int_{e^e}^{e^{e^{e}}}\left\{\ln (\ln (\ln x))+\frac{1}{(\ln x)\ln (\ln x)}\right\}dx.\] Own

Today's calculation of integrals, 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 $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

2007 Today's Calculation Of Integral, 175

Tags: calculus , integration , calculus computations , logarithm

Evaluate $\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.$

2010 Today's Calculation Of Integral, 558

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

For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$. Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$

2007 Today's Calculation Of Integral, 218

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

For any quadratic functions $ f(x)$ such that $ f'(2)\equal{}1$, evaluate $ \int_{2\minus{}\pi}^{2\plus{}\pi}f(x)\sin\left(\frac{x}{2}\minus{}1\right) dx$.

2009 Today's Calculation Of Integral, 516

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

Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2010 Today's Calculation Of Integral, 648

Tags: calculus , integration , function , calculus computations

Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that: (a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$ (b) For $x\neq 0,\ f(x)>0.$ Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer. $(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\] $(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\] [i]2010 Kyoto University, Master Course in Mathematics[/i]

2007 Today's Calculation Of Integral, 169

Tags: calculus , integration , function , calculus computations

(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$ (2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$ Find $a_{n},\ b_{n}.$

2009 Today's Calculation Of Integral, 512

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

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

2009 Today's Calculation Of Integral, 401

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

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

2007 Today's Calculation Of Integral, 206

Tags: calculus , integration , calculus computations

Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$

2010 Today's Calculation Of Integral, 530

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

Answer the following questions. (1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$. (2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$. (3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.

2009 Today's Calculation Of Integral, 457

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

Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$

2011 Today's Calculation Of Integral, 740

Tags: calculus , integration , geometry , derivative , calculus computations

Let $r$ be a positive constant. If 2 curves $C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}$ have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by $C_1,\ C_2$.