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, 198
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Compare the values of the following definite integrals. \[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]

2007 Today's Calculation Of Integral, 207
Tags: calculus , integration , calculus computations , logarithm
Evaluate the following definite integral. \[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]

2010 Today's Calculation Of Integral, 550
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.

2007 Today's Calculation Of Integral, 209
Tags: calculus , integration , trigonometry , calculus computations
Let $m,\ n$ be the given distinct positive integers. Answer the following questions. (1) Find the real number $\alpha \ (|\alpha |<1)$ such that $\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0$. (2) Find the real number $\beta$ satifying the sytem of equation $\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}$, $\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}$.

2010 Today's Calculation Of Integral, 644
Tags: calculus , integration , trigonometry , inequalities , calculus computations , logarithm
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that \[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\] Own

2010 Today's Calculation Of Integral, 585
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.

2012 Today's Calculation Of Integral, 816
Tags: calculus , integration , geometry , geometric transformation , rotation , combinatorial geometry , calculus computations
Find the volume of the solid of a circle $x^2+(y-1)^2=4$ generated by a rotation about the $x$-axis.

2011 Today's Calculation Of Integral, 720
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

2010 Today's Calculation Of Integral, 617
Tags: calculus , integration , function , calculus computations
Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

2010 Today's Calculation Of Integral, 595
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$ 2009 Kumamoto University entrance exam/Medicine

2011 Today's Calculation Of Integral, 678
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\] [i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]

2010 Today's Calculation Of Integral, 658
Tags: calculus , integration , trigonometry , geometry , analytic geometry , calculus computations
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$ (1) Find the length $L$ of $C$. (2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$. Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School. [i]1997 Kyoto University entrance exam/Science[/i]

2005 Today's Calculation Of Integral, 63
Tags: calculus , integration , limit , trigonometry , calculus computations
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$ Evaluate \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

2010 Today's Calculation Of Integral, 621
Tags: calculus , integration , limit , real analysis , calculus computations , logarithm
Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

2010 Today's Calculation Of Integral, 555
Tags: calculus , integration , derivative , function , calculus computations , logarithm
For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

2013 Today's Calculation Of Integral, 864
Tags: calculus , integration , inequalities , limit , calculus computations
Let $m,\ n$ be positive integer such that $2\leq m<n$. (1) Prove the inequality as follows. \[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\] (2) Prove the inequality as follows. \[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\] (3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows. \[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]

2007 Today's Calculation Of Integral, 224
Tags: calculus , integration , trigonometry , calculus computations
Let $ f(x)\equal{}x^{2}\plus{}|x|$. Prove that $ \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx$.

2010 Today's Calculation Of Integral, 573
Tags: calculus , integration , geometry , trigonometry , calculus computations , logarithm
Find the area of the figure bounded by three curves $ C_1: y\equal{}\sin x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_2: y\equal{}\cos x\ \left(0\leq x<\frac {\pi}{2}\right)$ $ C_3: y\equal{}\tan x\ \left(0\leq x<\frac {\pi}{2}\right)$.

2009 Today's Calculation Of Integral, 481
Tags: calculus , integration , trigonometry , calculus computations
For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$. Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$

2009 Today's Calculation Of Integral, 489
Tags: calculus , integration , limit , function , calculus computations
Find the following limit. $ \lim_{n\to\infty} \int_{\minus{}1}^1 |x|\left(1\plus{}x\plus{}\frac{x^2}{2}\plus{}\frac{x^3}{3}\plus{}\cdots \plus{}\frac{x^{2n}}{2n}\right)\ dx$.

Today's calculation of integrals, 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 $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$

Today's calculation of integrals, 890
Tags: calculus , integration , function , algorithm , calculus computations
A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

2007 Today's Calculation Of Integral, 234
Tags: calculus , integration , function , trigonometry , calculus computations
For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$. Evaluate $ \int_0^{100\pi } f(x)\ dx.$

Today's calculation of integrals, 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, 462
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^1 \frac{(1\minus{}x\plus{}x^2)\cos \ln (x\plus{}\sqrt{1\plus{}x^2})\minus{}\sqrt{1\plus{}x^2}\sin \ln (x\plus{}\sqrt{1\plus{}x^2})}{(1\plus{}x^2)^{\frac{3}{2}}}\ dx$.