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 $I(a)=\int_1^e |\ln ax|\ dx.$

2005 Today's Calculation Of Integral, 16
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Calculate the following indefinite integrals. [1] $\int \sin (\ln x)dx$ [2] $\int \frac{x+\sin ^ 2 x}{x\sin ^ 2 x}dx$ [3] $\int \frac{x^3}{x^2+1}dx$ [4] $\int \frac{x^2}{\sqrt{2x-1}}dx$ [5] $\int \frac{x+\cos 2x +1}{x\cos ^ 2 x}dx$

2009 Today's Calculation Of Integral, 447
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $ \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{x^2}{(1\plus{}x\tan x)(x\minus{}\tan x)\cos ^ 2 x}\ dx.$

Today's calculation of integrals, 866
Tags: calculus , integration , geometry , trigonometry , calculus computations
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

2014 Contests, 903
Tags: calculus , integration , function , real analysis , geometric series , calculus computations
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$. Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$

2009 Today's Calculation Of Integral, 432
Tags: calculus , integration , function , calculus computations , logarithm
Define the function $ f(t)\equal{}\int_0^1 (|e^x\minus{}t|\plus{}|e^{2x}\minus{}t|)dx$. Find the minimum value of $ f(t)$ for $ 1\leq t\leq e$.

2012 Today's Calculation Of Integral, 836
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

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]

2012 Today's Calculation Of Integral, 797
Tags: calculus , integration , geometry , 3d geometry , tetrahedron , trigonometry , calculus computations
In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$. Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$. 50 points

2011 Today's Calculation Of Integral, 706
Tags: calculus , integration , calculus computations
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that \[\left\{ \begin{array}{ll} x^2+y^2\leq 4 &\quad \\ 0\leq z\leq 4 &\quad \end{array} \right.\] Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying \[\left\{ \begin{array}{ll} z\leq (x-2)^2 &\quad \\ z\leq y^2 &\quad \end{array} \right.\] Find the volume of the solid $V$.

2009 Today's Calculation Of Integral, 464
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.

2012 ISI Entrance Examination, 2
Tags: function , limit , calculus , derivative , calculus computations
Consider the following function \[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\] [b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$ [b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.

2008 Harvard-MIT Mathematics Tournament, 1
Tags: algebra , polynomial , function , calculus , calculus computations
Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

2014 Contests, 2
Tags: calculus , integration , parabola , geometry , calculus computations , conic
Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.

2005 Today's Calculation Of Integral, 37
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate \[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]

2012 Today's Calculation Of Integral, 829
Tags: calculus , integration , calculus computations , logarithm
Let $a$ be a positive constant. Find the value of $\ln a$ such that \[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]

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]

2009 Today's Calculation Of Integral, 441
Tags: calculus , integration , calculus computations , logarithm
Evaluate $ \int_1^e \frac{(x^2\ln x\minus{}1)e^x}{x}\ dx.$

2013 Today's Calculation Of Integral, 895
Tags: calculus , integration , analytic geometry , parabola , geometry , calculus computations , conic
In the coordinate plane, suppose that the parabola $C: y=-\frac{p}{2}x^2+q\ (p>0,\ q>0)$ touches the circle with radius 1 centered on the origin at distinct two points. Find the minimum area of the figure enclosed by the part of $y\geq 0$ of $C$ and the $x$-axis.

2007 Today's Calculation Of Integral, 199
Tags: calculus , integration , calculus computations
Let $m,\ n$ be non negative integers. Calculate \[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \] where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.

2005 Today's Calculation Of Integral, 61
Tags: calculus , integration , trigonometry , calculus computations
Evaluate \[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]

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

2012 Today's Calculation Of Integral, 833
Tags: calculus , integration , trigonometry , derivative , complex number , calculus computations
Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

2009 Today's Calculation Of Integral, 406
Tags: calculus , integration , limit , trigonometry , calculus computations
Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

2007 Today's Calculation Of Integral, 247
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.$