Found problems: 713
2010 Today's Calculation Of Integral, 626
Find $\lim_{a\rightarrow +0} \int_a^1 \frac{x\ln x}{(1+x)^3}dx.$
[i]2010 Nara Medical University entrance exam[/i]
2007 Today's Calculation Of Integral, 171
Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$
2013 Waseda University Entrance Examination, 3
Let $f(x)=\frac 12e^{2x}+2e^x+x$. Answer the following questions.
(1) For a real number $t$, set $g(x)=tx-f(x).$ When $x$ moves in the range of all real numbers, find the range of $t$ for which $g(x)$ has maximum value, then for the range of $t$, find the maximum value of $g(x)$ and the value of $x$ which gives the maximum value.
(2) Denote by $m(t)$ the maximum value found in $(1)$. Let $a$ be a constant, consider a function of $t$, $h(t)=at-m(t)$. When $t$ moves in the range of $t$ found in $(1)$, find the maximum value of $h(t)$.
2010 Today's Calculation Of Integral, 568
Throw $ n$ balls in to $ 2n$ boxes.　Suppose each ball comes into each box with equal probability of entering in any boxes.
Let $ p_n$ be the probability such that any box has ball less than or equal to one. Find the limit $ \lim_{n\to\infty} \frac{\ln p_n}{n}$
2010 Today's Calculation Of Integral, 637
For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions:
(1) Calculate $I_{n+2}+I_n.$
(2) Evaluate the values of $I_1,\ I_2$ and $I_3.$
1978 Niigata university entrance exam
2011 Today's Calculation Of Integral, 760
Prove that there exists a positive integer $n$ such that $\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.$
2012 Today's Calculation Of Integral, 793
Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.
2010 Today's Calculation Of Integral, 529
Prove that the following inequality holds for each natural number $ n$.
\[ \int_0^{\frac {\pi}{2}} \sum_{k \equal{} 1}^n \left(\frac {\sin kx}{k}\right)^2dx < \frac {61}{144}\pi\]
2010 Today's Calculation Of Integral, 559
In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions.
(1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$.
(2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$.
(3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$.
Then use this to find the area of (2).
2010 Today's Calculation Of Integral, 638
Let $(a,\ b)$ be a point on the curve $y=\frac{x}{1+x}\ (x\geq 0).$ Denote $U$ the volume of the figure enclosed by the curve , the $x$ axis and the line $x=a$, revolved around the the $x$ axis and denote $V$ the volume of the figure enclosed by the curve , the $y$ axis and th line $y=b$, revolved around the $y$ axis. What's the relation of $U$ and $V?$
1978 Chuo university entrance exam/Science and Technology
2007 Today's Calculation Of Integral, 231
Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.
2007 Today's Calculation Of Integral, 233
Find the minimum value of the following definite integral.
$ \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.$
2011 Today's Calculation Of Integral, 732
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
2010 Today's Calculation Of Integral, 562
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
2010 Today's Calculation Of Integral, 661
Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$.
(1) Find the 36th term.
(2) Find $\int x^2\ln x\ dx$.
(3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part?
If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$.
[i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]
2012 Today's Calculation Of Integral, 830
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$
2011 Today's Calculation Of Integral, 691
Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$.
Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis.
[i]2011 Yokohama National Universty entrance exam/Engineering[/i]
2011 Today's Calculation Of Integral, 694
Prove the following inequality:
\[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\]
created by kunny
2012 Today's Calculation Of Integral, 779
Consider parabolas $C_a: y=-2x^2+4ax-2a^2+a+1$ and $C: y=x^2-2x$ in the coordinate plane.
When $C_a$ and $C$ have two intersection points, find the maximum area enclosed by these parabolas.
2009 Today's Calculation Of Integral, 493
In the $ x \minus{} y$ plane, let $ l$ be the tangent line at the point $ A\left(\frac {a}{2},\ \frac {\sqrt {3}}{2}b\right)$ on the ellipse $ \frac {x^2}{a^2} \plus{} \frac {y^2}{b^2}\equal{}1\ (0 < b < 1 < a)$. Let denote $ S$ be the area of the figure bounded by $ l,$ the $ x$ axis and the ellipse.
(1) Find the equation of $ l$.
(2) Express $ S$ in terms of $ a,\ b$.
(3) Find the maximum value of $ S$ with the constraint $ a^2 \plus{} 3b^2 \equal{} 4$.
2010 District Olympiad, 4
Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that:
\[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]
2010 Today's Calculation Of Integral, 579
Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$
2007 Today's Calculation Of Integral, 242
A cubic function $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ touches a line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha$ and intersects $ x \equal{} \beta \ (\alpha \neq \beta)$.
Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta$.
2005 Today's Calculation Of Integral, 74
$p,q$ satisfies $px+q\geq \ln x$ at $a\leq x\leq b\ (0<a<b)$.
Find the value of $p,q$ for which the following definite integral is minimized and then the minimum value.
\[\int_a^b (px+q-\ln x)dx\]
2011 Today's Calculation Of Integral, 763
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$