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

Tags were heavily modified to better represent problems.

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Found problems: 713

2009 Today's Calculation Of Integral, 519
Tags: calculus , integration , calculus computations
Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.

2011 Today's Calculation Of Integral, 731
Tags: calculus , integration , parabola , geometry , calculus computations , conic
Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively. When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.

2007 Today's Calculation Of Integral, 210
Tags: calculus , integration , trigonometry , function , calculus computations , logarithm
Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.

2007 Today's Calculation Of Integral, 186
Tags: calculus , integration , limit , calculus computations
For $a>0,$ find $\lim_{a\to\infty}a^{-\left(\frac{3}{2}+n\right) }\int_{0}^{a}x^{n}\sqrt{1+x}\ dx\ (n=1,\ 2,\ \cdots).$

2010 Today's Calculation Of Integral, 618
Tags: calculus , integration , trigonometry , calculus computations
Find the minimu value of $\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.$ [i]2010 Ibaraki University entrance exam/Science[/i]

2013 Today's Calculation Of Integral, 882
Tags: calculus , integration , limit , calculus computations , riemann sum , logarithm
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.

2009 Today's Calculation Of Integral, 430
Tags: calculus , integration , trigonometry , limit , calculus computations
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

2012 Today's Calculation Of Integral, 807
Tags: calculus , integration , limit , calculus computations
Define a sequence $a_n$ satisfying : \[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\] Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$

2011 Today's Calculation Of Integral, 767
Tags: calculus , integration , trigonometry , calculus computations
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$ Evaluate $\int_0^1 f(t)dt.$

2007 Today's Calculation Of Integral, 187
Tags: calculus , integration , trigonometry , inequalities , calculus computations
For a constant $a,$ let $f(x)=ax\sin x+x+\frac{\pi}{2}.$ Find the range of $a$ such that $\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).$

2010 Today's Calculation Of Integral, 594
Tags: calculus , integration , geometry , calculus computations
In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$. Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.

2009 Today's Calculation Of Integral, 411
Tags: calculus , integration , geometry , calculus computations
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.

2010 Today's Calculation Of Integral, 593
Tags: calculus , integration , trigonometry , function , calculus computations
For a positive integer $m$, prove the following ineqaulity. $0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$ 1996 Osaka University entrance exam

2005 Today's Calculation Of Integral, 23
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Evaluate \[\lim_{a\rightarrow \frac{\pi}{2}-0}\ \int_0^a\ (\cos x)\ln (\cos x)\ dx\ \left(0\leqq a <\frac{\pi}{2}\right)\]

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$.

2011 Today's Calculation Of Integral, 727
Tags: calculus , integration , geometry , limit , calculus computations , logarithm
For positive constant $a$, let $C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})$. Denote by $l(t)$ the length of the part $a\leq y\leq t$ for $C$ and denote by $S(t)$ the area of the part bounded by the line $y=t\ (a<t)$ and $C$. Find $\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.$

2012 Kyoto University Entry Examination, 1
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Answer the following questions: (1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

2010 Today's Calculation Of Integral, 588
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.

2010 Today's Calculation Of Integral, 645
Tags: calculus , integration , function , inequalities , calculus computations
Prove the following inequality. \[\int_{-1}^1 \frac{e^x+e^{-x}}{e^{e^{e^x}}}dx<e-\frac{1}{e}\] Own

Today's calculation of integrals, 849
Tags: calculus , integration , function , calculus computations
Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$

2012 Waseda University Entrance Examination, 4
Tags: calculus , integration , function , analytic geometry , calculus computations , logarithm
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions: (1) Find $f'(x)$. (2) Sketch the graph of $y=f(x)$. (3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.

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 $\lim_{n\to \infty} P_n$

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

2007 Today's Calculation Of Integral, 213
Tags: calculus , integration , calculus computations
Find the minimum value of $ f(a)=\int_{0}^{1}x|x-a|\ dx$.

2010 ISI B.Math Entrance Exam, 8
Tags: function , limit , calculus , calculus computations
Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.