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

2010 Today's Calculation Of Integral, 590
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac{\pi}{8}} \frac{(\cos \theta \plus{}\sin \theta)^{\frac{3}{2}}\minus{}(\cos \theta \minus{}\sin \theta)^{\frac{3}{2}}}{\sqrt{\cos 2\theta}}\ d\theta$.

2005 Today's Calculation Of Integral, 38
Tags: calculus , integration , calculus computations , logarithm
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

2012 Today's Calculation Of Integral, 820
Tags: calculus , integration , analytic geometry , limit , calculus computations , logarithm
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$

2012 Today's Calculation Of Integral, 806
Tags: calculus , integration , trigonometry , calculus computations
Let $n$ be positive integers and $t$ be a positive real number. Evaluate $\int_0^{\frac{2n}{t}\pi} |x\sin\ tx|\ dx.$

2005 Today's Calculation Of Integral, 53
Tags: calculus , integration , trigonometry , calculus computations
Find the maximum value of the following integral. \[\int_0^{\infty} e^{-x}\sin tx\ dx\]

2009 Today's Calculation Of Integral, 459
Tags: calculus , integration , limit , calculus computations , logarithm
Find $ \lim_{x\to\infty} \int_{e^{\minus{}x}}^1 \left(\ln \frac{1}{t}\right)^ n\ dt\ (x\geq 0,\ n\equal{}1,\ 2,\ \cdots)$.

2005 Today's Calculation Of Integral, 51
Tags: calculus , integration , function , calculus computations
A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$. If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$

2005 Today's Calculation Of Integral, 29
Tags: calculus , integration , trigonometry , calculus computations
Let $a$ be a real number. Evaluate \[\int _{-\pi+a}^{3\pi+a} |x-a-\pi|\sin \left(\frac{x}{2}\right)dx\]

2009 Today's Calculation Of Integral, 416
Tags: calculus , integration , trigonometry , calculus computations
Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

2008 ISI B.Math Entrance Exam, 1
Tags: calculus , integration , function , calculus computations
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function . Suppose \[f(x)=\frac{1}{t} \int^t_0 (f(x+y)-f(y))\,dy\] $\forall x\in \mathbb{R}$ and all $t>0$ . Then show that there exists a constant $c$ such that $f(x)=cx\ \forall x$

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

2012 Today's Calculation Of Integral, 835
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate the following definite integrals. (a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$ (b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$ (c) $\int_1^e x\ln \sqrt{x}\ dx$ (d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$

2005 Today's Calculation Of Integral, 44
Tags: calculus , integration , trigonometry , induction , abstract algebra , calculus computations
Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

2007 Today's Calculation Of Integral, 221
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.

2009 Today's Calculation Of Integral, 471
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac{1\minus{}x(e^x\minus{}1)}{x(1\plus{}xe^x\ln x)}\ dx$.

2010 Today's Calculation Of Integral, 580
Tags: calculus , integration , analytic geometry , geometry , calculus computations
Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$.　Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

2007 Today's Calculation Of Integral, 174
Tags: calculus , integration , parameterization , calculus computations , geometry
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 $C.$

2010 Today's Calculation Of Integral, 643
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

2012 Today's Calculation Of Integral, 771
Tags: calculus , integration , parabola , geometry , analytic geometry , calculus computations , conic
(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis. (2) For the range of $a$ found in the previous question, express the area bounded by the two tangent lines and the parabola $y=(x+a)^2$ in terms of $a$.

2005 Today's Calculation Of Integral, 91
Tags: calculus , integration , calculus computations
Prove the following inequality. \[ \sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005} \]

2005 Today's Calculation Of Integral, 77
Tags: calculus , integration , geometry , calculus computations
Find the area of the part enclosed by the following curve. \[x^2+2axy+y^2=1\ (-1<a<1)\]

2007 Today's Calculation Of Integral, 184
Tags: calculus , integration , function , inequalities , real analysis , calculus computations , logarithm
(1) For real numbers $x,\ a$ such that $0<x<a,$ prove the following inequality. \[\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right). \] (2) Use the result of $(1)$ to prove that $0.68<\ln 2<0.71.$

2010 Today's Calculation Of Integral, 652
Tags: calculus , integration , calculus computations
Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$ Evaluate \[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\] [i]1981 Tokyo University, Master Course[/i]

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]

2013 Today's Calculation Of Integral, 879
Tags: calculus , integration , trigonometry , calculus computations , integral
Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$