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

2012 Today's Calculation Of Integral, 804
Tags: calculus , integration , derivative , function , calculus computations , logarithm
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$

2009 Today's Calculation Of Integral, 437
Tags: calculus , integration , calculus computations
Evaluate $ \int_0^1 \frac{1}{\sqrt{x}\sqrt{1\plus{}\sqrt{x}}\sqrt{1\plus{}\sqrt{1\plus{}\sqrt{x}}}}\ dx.$

2010 Today's Calculation Of Integral, 527
Tags: calculus , integration , calculus computations
Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

2005 Today's Calculation Of Integral, 24
Tags: calculus , integration , trigonometry , calculus computations
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.

2009 Today's Calculation Of Integral, 433
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx$.

2011 Today's Calculation Of Integral, 752
Tags: calculus , integration , induction , calculus computations
Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$

2011 Tokyo Instutute Of Technology Entrance Examination, 2
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]

2010 Today's Calculation Of Integral, 625
Tags: calculus , integration , limit , trigonometry , calculus computations
Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

2005 Today's Calculation Of Integral, 42
Tags: calculus , integration , limit , trigonometry , calculus computations , logarithm
Let $0<t<\frac{\pi}{2}$. Evaluate \[\lim_{t\rightarrow \frac{\pi}{2}} \int_0^t \tan \theta \sqrt{\cos \theta}\ln (\cos \theta)d\theta\]

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}$.

2010 Today's Calculation Of Integral, 535
Tags: calculus , integration , trigonometry , geometry , geometric transformation , rotation , calculus computations
Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$, $ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\ y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$ When the point $ P$ moves on the curve $ C$, (1) Find the magnitude of acceleralation of the point $ P$ at time $ t$. (2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$. (3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis. Edited.

2010 Today's Calculation Of Integral, 631
Tags: calculus , integration , calculus computations , logarithm
Evaluate $\int_{\sqrt{2}}^{\sqrt{3}} (x^2+\sqrt{x^4-1})(\frac{1}{\sqrt{x^2+1}}+{\frac{1}{\sqrt{x^2-1}})dx.}$ [i]Proposed by kunny[/i]

2010 Today's Calculation Of Integral, 537
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

2005 Today's Calculation Of Integral, 35
Tags: calculus , integration , derivative , function , calculus computations
Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

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 $0.68<\ln 2<0.71.$

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 \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

2005 Today's Calculation Of Integral, 72
Tags: calculus , integration , function , trigonometry , calculus computations
Let $f(x)$ be a continuous function satisfying $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$ Find the value of $k$ for which $\int_0^{\pi} f(x)dx$ is maximized.

2012 Today's Calculation Of Integral, 788
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 $Q$.

2007 Today's Calculation Of Integral, 239
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$

2009 Today's Calculation Of Integral, 516
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Let $ f(x)\equal{}\frac{1}{\sin x\sqrt{1\minus{}\cos x}}\ (0<x<\pi)$. (1) Find the local minimum value of $ f(x)$. (2) Evaluate $ \int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} f(x)\ dx$.

2007 Today's Calculation Of Integral, 254
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx.$ Sorry, I have deleted my first post because that was wrong. kunny

2010 Today's Calculation Of Integral, 591
Tags: calculus , integration , inequalities , calculus computations
Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$. Prove the following inequality: \[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]

2005 Today's Calculation Of Integral, 17
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

2005 Today's Calculation Of Integral, 27
Tags: calculus , integration , trigonometry , quadratic , algebra , calculus computations
Let $f(x)=t\sin x+(1-t)\cos x\ (0\leqq t\leqq 1)$. Find the maximum and minimum value of the following $P(t)$. \[P(t)=\left\{\int_0^{\frac{\pi}{2}} e^x f(x) dx \right\}\left\{\int_0^{\frac{\pi}{2}} e^{-x} f(x)dx \right\}\]

2012 Today's Calculation Of Integral, 810
Tags: calculus , integration , function , calculus computations
Given the functions $f(x)=xe^{x}+2x\int_0^2 |g(t)|dt-1,\ g(x)=x^2-x\int_0^1 f(t)dt$, evaluate $\int_0^2 |g(t)|dt.$