This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 713

2005 Today's Calculation Of Integral, 4
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \frac{x}{\sqrt{5-x}}dx$ [2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$ [3] $\int (\sin x+\cos x)^2dx$ [4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$ [5]$\int (\sin x+\sin 2x)^2 dx$

2010 Today's Calculation Of Integral, 546
Tags: calculus , integration , trigonometry , derivative , calculus computations
Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.

2010 Today's Calculation Of Integral, 636
Tags: calculus , integration , limit , derivative , calculus computations , logarithm , geometry
Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$ (1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$. (2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$. [i]1992 Tokyo University entrance exam/Science[/i]

2007 IMS, 5
Tags: limit , calculus , calculus computations
Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]

2010 Today's Calculation Of Integral, 615
Tags: calculus , integration , derivative , calculus computations , logarithm
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2011 Today's Calculation Of Integral, 729
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$

2008 Harvard-MIT Mathematics Tournament, 3
Tags: calculus , integration , calculus computations , logarithm
([b]4[/b]) Find all $ y > 1$ satisfying $ \int^y_1x\ln x\ dx \equal{} \frac {1}{4}$.

2009 Today's Calculation Of Integral, 479
Tags: calculus , integration , trigonometry , calculus computations
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral: $ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$

2007 Today's Calculation Of Integral, 188
Tags: calculus , integration , calculus computations
Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.

Today's calculation of integrals, 848
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2007 Today's Calculation Of Integral, 237
Tags: calculus , integration , absolute value , calculus computations , logarithm
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$

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 $m$.

2013 Today's Calculation Of Integral, 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 $\sum_{n=1}^{\infty} I_n.$

Today's calculation of integrals, 885
Tags: calculus , integration , trigonometry , calculus computations , indefinite integral , logarithm
Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

2010 Contests, 524
Tags: calculus , integration , function , calculus computations
Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

2013 Today's Calculation Of Integral, 890
Tags: calculus , integration , function , algorithm , calculus computations
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

2005 Today's Calculation Of Integral, 21
Tags: calculus , integration , trigonometry , calculus computations , logarithm
[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

2009 Today's Calculation Of Integral, 511
Tags: calculus , integration , derivative , trigonometry , calculus computations
Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

2010 Today's Calculation Of Integral, 540
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.

2013 Today's Calculation Of Integral, 885
Tags: calculus , integration , trigonometry , calculus computations , indefinite integral , logarithm
Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

Today's calculation of integrals, 766
Tags: calculus , integration , function , trigonometry , inequalities , calculus computations
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2005 ISI B.Math Entrance Exam, 6
Tags: algebra , polynomial , integration , calculus , geometry , calculus computations
Let $a_0=0<a_1<a_2<...<a_n$ be real numbers . Supppose $p(t)$ is a real valued polynomial of degree $n$ such that $\int_{a_j}^{a_{j+1}} p(t)\,dt = 0\ \ \forall \ 0\le j\le n-1$ Show that , for $0\le j\le n-1$ , the polynomial $p(t)$ has exactly one root in the interval $ (a_j,a_{j+1})$

2001 Vietnam National Olympiad, 3
Tags: trigonometry , integration , calculus , calculus computations
For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

Today's calculation of integrals, 894
Tags: calculus , integration , calculus computations , logarithm , geometry
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$

1994 Vietnam National Olympiad, 3
Tags: trigonometry , calculus , calculus computations
Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.