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

2007 Today's Calculation Of Integral, 172
Tags: calculus , integration , trigonometry , euler , calculus computations
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$

2009 Today's Calculation Of Integral, 463
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{e^{\frac{1}{\cos ^ 2 x}}\sin x}{\cos ^ 3 x}\ dx$.

2007 Today's Calculation Of Integral, 229
Tags: calculus , integration , limit , trigonometry , function , calculus computations
Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.

2007 Today's Calculation Of Integral, 236
Tags: calculus , integration , trigonometry , calculus computations
Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$. \[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\]. [i]1998 Shinsyu University entrance exam/Textile Science[/i]

2007 Today's Calculation Of Integral, 242
Tags: calculus , integration , function , geometry , calculus computations
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$. Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.

2010 Today's Calculation Of Integral, 647
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

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

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, 486
Tags: calculus , integration , trigonometry , analytic geometry , geometry , calculus computations
Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$ Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions. (1) Express the coordiante of $ H$ in terms of $ t$. (2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$. (3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.

2010 Today's Calculation Of Integral, 623
Tags: calculus , integration , function , calculus computations
Find the continuous function satisfying the following equation. \[\int_0^x f(t)dt+\int_0^x tf(x-t)dt=e^{-x}-1.\] [i]1978 Shibaura Institute of Technology entrance exam[/i]

2011 Today's Calculation Of Integral, 700
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

2007 Today's Calculation Of Integral, 253
Tags: calculus , integration , induction , calculus computations
Evaluate $ \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.$

2011 Today's Calculation Of Integral, 723
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$

2005 Today's Calculation Of Integral, 80
Tags: calculus , integration , algebra , function , domain , ratio , calculus computations
Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

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]$\int (\sin x+\sin 2x)^2 dx$

2012 Today's Calculation Of Integral, 855
Tags: calculus , integration , function , limit , calculus computations
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$. For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

2009 Today's Calculation Of Integral, 424
Tags: calculus , integration , geometry , limit , trigonometry , calculus computations
Let $ n$ be positive integer. For $ n \equal{} 1,\ 2,\ 3,\ \cdots n$, let denote $ S_k$ be the area of $ \triangle{AOB_k}$ such that $ \angle{AOB_k} \equal{} \frac {k}{2n}\pi ,\ OA \equal{} 1,\ OB_k \equal{} k$. Find the limit $ \lim_{n\to\infty}\frac {1}{n^2}\sum_{k \equal{} 1}^n S_k$.

2010 Today's Calculation Of Integral, 572
Tags: calculus , integration , trigonometry , logarithm , calculus computations
For integer $ n,\ a_n$ is difined by $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx$. (1) Find $ a_{\minus{}2},\ a_{\minus{}1}$. (2) Find the relation of $ a_n$ and $ a_{n\minus{}2}$. (3) Prove that $ a_{2n}\equal{}b_n\plus{}\pi c_n$ for some rational number $ b_n,\ c_n$, then find $ c_n$ for $ n<0$.

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

2012 Today's Calculation Of Integral, 776
Tags: calculus , integration , calculus computations
Evaluate $\int_{\frac{1-\sqrt{5}}{2}}^{\frac{1+\sqrt{5}}{2}} (2x^2-1)e^{2x}dx.$

Today's calculation of integrals, 872
Tags: calculus , integration , trigonometry , geometry , limit , function , calculus computations
Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

2005 ISI B.Math Entrance Exam, 2
Tags: limit , induction , calculus , calculus computations
Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define : $P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$ Compute $\lim_{n\to \infty} P_n$

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}$.

2005 Today's Calculation Of Integral, 48
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Evaluate \[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]

2009 Today's Calculation Of Integral, 451
Tags: calculus , integration , limit , calculus computations , logarithm
Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).$