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, 609
Tags: calculus , integration , function , trigonometry , calculus computations
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number. 1972 Tokyo University of Education entrance exam

2012 Today's Calculation Of Integral, 848
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$

2012 Today's Calculation Of Integral, 839
Tags: calculus , integration , geometry , calculus computations
Evaluate $\int_{\frac 12}^1 \sqrt{1-x^2}\ dx.$

Today's calculation of integrals, 881
Tags: calculus , integration , trigonometry , calculus computations , definite integral
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.

2009 Today's Calculation Of Integral, 442
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

2011 Today's Calculation Of Integral, 688
Tags: calculus , integration , trigonometry , calculus computations , logarithm
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

2005 Today's Calculation Of Integral, 26
Tags: calculus , integration , calculus computations , logarithm
Evaluate \[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]

2009 Today's Calculation Of Integral, 517
Tags: calculus , integration , geometry , search , calculus computations
Consider points $ P$ which are inside the square with side length $ a$ such that the distance from $ P$ to the center of the square equals to the least distance from $ P$ to each side of the square.Find the area of the figure formed by the whole points $ P$.

2005 Today's Calculation Of Integral, 22
Tags: calculus , integration , induction , trigonometry , calculus computations
Evaluate \[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]

2013 Today's Calculation Of Integral, 899
Tags: calculus , integration , limit , calculus computations
Find the limit as below. \[\lim_{n\to\infty} \frac{(1^2+2^2+\cdots +n^2)(1^3+2^3+\cdots +n^3)(1^4+2^4+\cdots +n^4)}{(1^5+2^5+\cdots +n^5)^2}\]

2011 Today's Calculation Of Integral, 676
Tags: calculus , integration , trigonometry , calculus computations
Let $f(x)=\cos ^ 4 x+3\sin ^ 4 x$. Evaluate $\int_0^{\frac{\pi}{2}} |f'(x)|dx$. [i]2011 Tokyo University of Science entrance exam/Management[/i]

2011 Today's Calculation Of Integral, 746
Tags: calculus , integration , floor function , inequalities , calculus computations
Prove the following inequality. \[n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.\]

2007 Today's Calculation Of Integral, 223
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.

2007 Today's Calculation Of Integral, 185
Tags: calculus , integration , trigonometry , calculus computations
Evaluate the following integrals. (1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$ (2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$

2011 Today's Calculation Of Integral, 716
Tags: calculus , integration , calculus computations , logarithm
Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}\]

2007 Today's Calculation Of Integral, 226
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx$ [color=darkblue]Virgil Nicula have already posted the integral[/color] :oops:

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

2008 Harvard-MIT Mathematics Tournament, 6
Tags: limit , ratio , calculus , calculus computations
Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.

2010 Today's Calculation Of Integral, 577
Tags: calculus , integration , inequalities , calculus computations
Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

2011 Today's Calculation Of Integral, 745
Tags: calculus , integration , trigonometry , calculus computations
When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$

2008 Teodor Topan, 3
Tags: limit , geometry , geometric transformation , calculus , calculus computations , logarithm
Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.

2009 Today's Calculation Of Integral, 494
Tags: calculus , integration , hyperbola , analytic geometry , calculus computations , conic
Suppose the curve $ C: y \equal{} ax^3 \plus{} 4x\ (a\neq 0)$ has a common tangent line at the point $ P$ with the hyperbola $ xy \equal{} 1$ in the first quadrant. (1) Find the value of $ a$ and the coordinate of the point $ P$. (2) Find the volume formed by the revolution of the solid of the figure bounded by the line segment $ OP$ and the curve $ C$ about the line $ OP$. [color=green][Edited.][/color]

2010 Today's Calculation Of Integral, 630
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$

2011 Today's Calculation Of Integral, 715
Tags: calculus , integration , function , calculus computations
Find the differentiable function $f(x)$ with $f(0)\neq 0$ satisfying $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all real numbers $x,\ y$.

2011 Today's Calculation Of Integral, 756
Tags: calculus , integration , geometry , analytic geometry , calculus computations
Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$ Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.