Found problems: 713
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
If a positive sequence $\{a_n\}_{n\geq 1}$ satisfies $\int_0^{a_n} x^{n}\ dx=2$, then find $\lim_{n\to\infty} a_n.$
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
For a positive integer $m$, prove the following ineqaulity.
$0\leq \int_0^1 \left(x+1-\sqrt{x^2+2x\cos \frac{2\pi}{2m+1}+1\right)dx\leq 1.}$
1996 Osaka University entrance exam
Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$
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]
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}$.
Calculate the following integarls.
[1] $\int x\cos ^ 2 x dx$
[2] $\int \frac{x-1}{(3x-1)^2}dx$
[3] $\int \frac{x^3}{(2-x^2)^4}dx$
[4] $\int \left({\frac{1}{4\sqrt{x}}+\frac{1}{2x}}\right)dx$
[5] $\int (\ln x)^2 dx$
Evaluate
\[\int_0^1 x^{2005}e^{-x^2}dx\]
Calculate the following indefinite integrals.
[1] $\int \tan ^ 3 x dx$
[2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$
[3] $\int \cos ^ 5 x dx$
[4] $\int \sin ^ 2 x\cos ^ 3 x dx$
[5]$ \int \frac{dx}{\sin x}$
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx$
[color=darkblue]Virgil Nicula have already posted the integral[/color] :oops:
Find the area of the part enclosed by the following curve.
\[x^2+2axy+y^2=1\ (-1<a<1)\]
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.
Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.
Let $a\geq 0$ be constant. Find the number of Intersection points of the graph of the function $y=x^3-3a^2x$ and the figure expressed by the equation $|x|+|y|=2$.
Find $f_n(x)$ such that $f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).$
Evaluate
\[\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).\]
[i]2011 Doshisya University entrance exam/Life Medical Sciences[/i]
Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$
Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$.
created by kunny
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$.
Let $f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.$
Find $\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.$
Prove that for each positive integer $n$
\[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]
Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$
Prove the following inequalities:
(1) For $0\leq x\leq 1$,
\[1-\frac 13x\leq \frac{1}{\sqrt{1+x^2}}\leq 1.\]
(2) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$
Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.