Found problems: 713
2010 Today's Calculation Of Integral, 617
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]
2007 Today's Calculation Of Integral, 232
For $ f(x)\equal{}1\minus{}\sin x$, let $ g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt.$
Show that $ g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x)$ for any real numbers $ x,\ y.$
2009 Today's Calculation Of Integral, 470
Determin integers $ m,\ n\ (m>n>0)$ for which the area of the region bounded by the curve $ y\equal{}x^2\minus{}x$ and the lines $ y\equal{}mx,\ y\equal{}nx$ is $ \frac{37}{6}$.
2009 Today's Calculation Of Integral, 479
Let $ a,\ b$ be real constants. Find the minimum value of the definite integral:
$ I(a,\ b)\equal{}\int_0^{\pi} (1\minus{}a\sin x \minus{}b\sin 2x)^2 dx.$
2010 Today's Calculation Of Integral, 628
(1) Evaluate the following definite integrals.
(a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$
(b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$
(c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$
(2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that
\[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\]
, then find the volume of the solid.
[i]1984 Yamanashi Medical University entrance exam[/i]
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny
2011 Today's Calculation Of Integral, 723
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$
2010 Today's Calculation Of Integral, 556
Prove the following inequality.
\[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\]
Last Edited.
Sorry, I have changed the problem.
kunny
2011 Today's Calculation Of Integral, 679
Find $\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}$.
[i]2011 Hosei University entrance exam/Design and Enginerring[/i]
2009 Today's Calculation Of Integral, 508
Compare the size of the definite integrals?
\[ \int_0^{\frac {\pi}{4}} x^{2008}\tan ^{2008}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2009}\tan ^{2009}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2010}\tan ^{2010}x\ dx\]
2011 Today's Calculation Of Integral, 738
Answer the following questions:
(1) Find the value of $a$ for which $S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx$ is minimized, then find the minimum value.
(2) Find the vlues of $p,\ q$ for which $T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx$ is minimized, then find the minimum value.
2001 Vietnam National Olympiad, 3
For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.
2005 Today's Calculation Of Integral, 1
Calculate the following indefinite integral.
[1] $\int \frac{e^{2x}}{(e^x+1)^2}dx$
[2] $\int \sin x\cos 3x dx$
[3] $\int \sin 2x\sin 3x dx$
[4] $\int \frac{dx}{4x^2-12x+9}$
[5] $\int \cos ^4 x dx$
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]
2010 Today's Calculation Of Integral, 616
Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$.
[i]2010 Hirosaki University entrance exam[/i]
2007 Today's Calculation Of Integral, 223
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.
2008 Harvard-MIT Mathematics Tournament, 8
Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.
2010 Today's Calculation Of Integral, 538
Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ 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 $h(t)$.
2011 Today's Calculation Of Integral, 722
Find the continuous function $f(x)$ such that :
\[\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12\]
2013 Today's Calculation Of Integral, 895
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.
2009 ISI B.Stat Entrance Exam, 6
Let $f(x)$ be a function satisfying
\[xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0\]
Show that $f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)$ where $f^{(n)}(x)$ denotes the $n$-th derivative evaluated at $x$.
2009 Today's Calculation Of Integral, 415
For a function $ f(x) \equal{} 6x(1 \minus{} x)$, suppose that positive constant $ c$ and a linear function $ g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0)$ satisfy the following 3 conditions: $ c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0$. Answer the following questions.
(1) Find the constants $ a,\ b,\ c$.
(2) For natural number $ n$, let $ I_n \equal{} \int_0^1 x^ne^x\ dx$. Express $ I_{n \plus{} 1}$ in terms of $ I_n$. Then evaluate $ I_1,\ I_2,\ I_3$.
(3) Evaluate the definite integrals $ \int_0^1 e^xf(x)\ dx$ and $ \int_0^1 e^xf(x)g(x)\ dx$.
(4) For real numbers $ s,\ t$, define $ J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx$. Find the constants $ A,\ B,\ C,\ D,\ E$ by setting $ J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F$.
(You don't need to find the constant $ F$).
(5) Find the values of $ s,\ t$ for which $ J$ is minimal.
2010 Today's Calculation Of Integral, 598
For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$.
(1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$.
(2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis.
(3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$.
1992 Tokyo University entrance exam/Science, 2nd exam
2011 Tokyo Instutute Of Technology Entrance Examination, 2
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]
Today's calculation of integrals, 889
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.