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

2011 Today's Calculation Of Integral, 735
Tags: trigonometry , logarithm , calculus computations , calculus , integration
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) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$

2009 Today's Calculation Of Integral, 498
Tags: calculus computations , integration , function , calculus
Let $ f(x)$ be a continuous function defined in the interval $ 0\leq x\leq 1.$ Prove that $ \int_0^1 xf(x)f(1\minus{}x)\ dx\leq \frac{1}{4}\int_0^1 \{f(x)^2\plus{}f(1\minus{}x)^2\}\ dx.$

Today's calculation of integrals, 859
Tags: geometry , calculus , derivative , integration , 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 $e^3>20.$

2007 Today's Calculation Of Integral, 180
Tags: trigonometry , integration , limit , calculus , geometry , calculus computations
Let $a_{n}$ be the area surrounded by the curves $y=e^{-x}$ and the part of $y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).$ Evaluate $\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).$

2009 Today's Calculation Of Integral, 480
Tags: integration , inequalities , calculus , calculus computations , geometry
Let $ a,\ b$ be positive real numbers. Prove that $ \int_{a \minus{} 2b}^{2a \minus{} b} \left|\sqrt {3b(2a \minus{} b) \plus{} 2(a \minus{} 2b)x \minus{} x^2} \minus{} \sqrt {3a(2b \minus{} a) \plus{} 2(2a \minus{} b)x \minus{} x^2}\right|dx$ $ \leq \frac {\pi}3 (a^2 \plus{} b^2).$ [color=green]Edited by moderator.[/color]

2010 Today's Calculation Of Integral, 571
Tags: function , logarithm , calculus , calculus computations , trigonometry , integration
Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.

2012 Hitotsubashi University Entrance Examination, 2
Tags: function , calculus , calculus computations
Let $a\geq 0$ be constant. Find the number of Intersection points of the graph of the function $y=x^3-3a^2x$ and the figure expressed by the equation $|x|+|y|=2$.

2010 Today's Calculation Of Integral, 613
Tags: calculus computations , geometry , integration , calculus , logarithm
Find the area of the part, in the $x$-$y$ plane, enclosed by the curve $|ye^{2x}-6e^{x}-8|=-(e^{x}-2)(e^{x}-4).$ [i]2010 Tokyo University of Agriculture and Technology entrance exam[/i]

2011 Today's Calculation Of Integral, 682
Tags: calculus computations , function , calculus , integration , trigonometry , geometry
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]

2013 Today's Calculation Of Integral, 898
Tags: integration , logarithm , calculus , calculus computations
Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

2010 Today's Calculation Of Integral, 636
Tags: calculus , integration , logarithm , derivative , geometry , limit , calculus computations
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]

2014 Contests, 901
Tags: calculus computations , integration , algebra , calculus , polynomial
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 $\int_0^1 P(x)dx=-\frac{52}{5}.$

2009 Today's Calculation Of Integral, 444
Tags: calculus computations , logarithm , integration , trigonometry , calculus
Evaluate $ \int_0^{\frac {\pi}{6}} \frac {\sin x \plus{} \cos x}{1 \minus{} \sin 2x}\ln\ (2 \plus{} \sin 2x)\ dx.$

2010 Today's Calculation Of Integral, 624
Tags: real analysis , integration , calculus computations , calculus , trigonometry , function
Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

2013 Today's Calculation Of Integral, 889
Tags: calculus computations , logarithm , integration , calculus , geometry
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

2013 Today's Calculation Of Integral, 874
Tags: conic , parabola , calculus , integration , geometry , calculus computations
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$. (1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$. (2) If $q=p+1$, then find the minimum value of $S$. (3) If $pq=-1$, then find the minimum value of $S$.

2014 Contests, 902
Tags: integration , calculus computations , calculus
For $a\geq 0$, find the minimum value of $S(a)=\int_0^1 |x^2+2ax+a^2-1|\ dx.$

2009 Today's Calculation Of Integral, 431
Tags: function , integration , calculus , trigonometry , calculus computations
Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$. (1) Find the maximum and minimum values of $ f(\theta)$. (2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.

2009 Today's Calculation Of Integral, 445
Tags: logarithm , integration , calculus , calculus computations
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.$

2009 Today's Calculation Of Integral, 513
Tags: integration , calculus computations , function , trigonometry , calculus
Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

2007 Today's Calculation Of Integral, 205
Tags: logarithm , integration , calculus , calculus computations
Evaluate the following definite integral. \[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]

2010 Today's Calculation Of Integral, 575
Tags: logarithm , integration , function , calculus computations , calculus , trigonometry
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions. (1) Find $ f'(x)$. (2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.

2009 Today's Calculation Of Integral, 398
Tags: geometry , search , calculus computations , calculus , integration , inequalities
In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality: $ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$ $ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$

2009 Today's Calculation Of Integral, 407
Tags: calculus , calculus computations , integration
Evaluate $ \int_0^1 (x \plus{} 3)\sqrt {xe^x}\ dx$.

2005 Today's Calculation Of Integral, 49
Tags: geometry , calculus , function , calculus computations , integration , trigonometry
For $x\geq 0$, Prove that $\int_0^x (t-t^2)\sin ^{2002} t \,dt<\frac{1}{2004\cdot 2005}$