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, 221
Tags: calculus computations , integration , logarithm , calculus
Evaluate $ \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx$.

2009 Today's Calculation Of Integral, 472
Tags: calculus computations , conic , integration , geometry , parabola , analytic geometry , calculus
Given a line segment $ PQ$ moving on the parabola $ y \equal{} x^2$ with end points on the parabola. The area of the figure surrounded by $ PQ$ and the parabola is always equal to $ \frac {4}{3}$. Find the equation of the locus of the mid point $ M$ of $ PQ$.

2007 Today's Calculation Of Integral, 230
Tags: calculus , logarithm , calculus computations , integration , function
Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.

2009 Today's Calculation Of Integral, 424
Tags: calculus , limit , integration , trigonometry , calculus computations , geometry
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$.

2005 Today's Calculation Of Integral, 65
Tags: integration , calculus , calculus computations
Let $a>0$. Find the minimum value of $\int_{-1}^a \left(1-\frac{x}{a}\right)\sqrt{1+x}\ dx$

2011 Today's Calculation Of Integral, 748
Tags: integration , calculus , logarithm , trigonometry , calculus computations
Evaluate the following integrals. (1) $\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).$ (2) $\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.$

2013 Today's Calculation Of Integral, 893
Tags: function , logarithm , derivative , integration , calculus , trigonometry , calculus computations
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$

2010 Today's Calculation Of Integral, 669
Tags: function , integration , calculus , calculus computations
Find the differentiable function defined in $x>0$ such that ${\int_1^{f(x)} f^{-1}(t)dt=\frac 13(x^{\frac {3}{2}}-8}).$

2007 Today's Calculation Of Integral, 243
Tags: calculus , geometry , function , calculus computations , integration
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.

2010 Today's Calculation Of Integral, 595
Tags: integration , trigonometry , logarithm , calculus , calculus computations
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

2009 Today's Calculation Of Integral, 509
Tags: algebra , calculus , integration , partial fractions , calculus computations , logarithm , trigonometry
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\tan x}{1\plus{}\sin x}\ dx$.

2010 Today's Calculation Of Integral, 614
Tags: calculus , integration , ratio , trigonometry , calculus computations
Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

2011 Today's Calculation Of Integral, 710
Tags: calculus computations , logarithm , trigonometry , calculus , integration
Evaluate $\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta$.

2012 Kyoto University Entry Examination, 1
Tags: logarithm , calculus , trigonometry , integration , limit , calculus computations
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

2007 Today's Calculation Of Integral, 245
Tags: calculus , integration , geometry , function , calculus computations
A sextic funtion $ y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma .$ created by kunny

2005 Today's Calculation Of Integral, 47
Tags: integration , function , trigonometry , calculus , calculus computations
Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$. \[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]

2011 Today's Calculation Of Integral, 730
Tags: logarithm , calculus computations , integration , limit , calculus
Let $a_n$ be the local maximum of $f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)$ for $x>0$. Find $\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}$.

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

2012 Today's Calculation Of Integral, 797
Tags: trigonometry , calculus computations , geometry , tetrahedron , integration , 3d geometry , calculus
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

2005 Today's Calculation Of Integral, 91
Tags: calculus computations , calculus , integration
Prove the following inequality. \[ \sum_{n=0}^\infty \int_0^1 x^{4011} (1-x^{2006})^\frac{n-1}{2006}\ dx<\frac{2006}{2005} \]

2010 Today's Calculation Of Integral, 615
Tags: calculus computations , derivative , logarithm , calculus , integration
For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

2012 Today's Calculation Of Integral, 841
Tags: calculus computations , integration , trigonometry , calculus , function
Find $\int_0^x \frac{dt}{1+t^2}+\int_0^{\frac{1}{x}} \frac{dt}{1+t^2}\ (x>0).$

2007 Today's Calculation Of Integral, 253
Tags: calculus computations , integration , calculus , induction
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.$

Today's calculation of integrals, 861
Tags: analytic geometry , calculus , calculus computations , integration , function , geometry
Answer the questions as below. (1) Find the local minimum of $y=x(1-x^2)e^{x^2}.$ (2) Find the total area of the part bounded the graph of the function in (1) and the $x$-axis.

2009 Today's Calculation Of Integral, 461
Tags: calculus computations , inequalities , calculus , integration , limit , logarithm
Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.