Found problems: 713
Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$
Define a sequence $a_n$ satisfying :
\[a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).\]
Find $\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.$
Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.
A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$
Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$
For $a>0$, find the minimum value of $I(a)=\int_1^e |\ln ax|\ dx.$
Find the minimum value of $\int_0^{\frac{\pi}{2}} |\cos x -a|\sin x \ dx$
Evaluate $ \int_0^{\pi} \frac{x\sin ^ 3 x}{\sin ^ 2 x\plus{}8}dx$.
Consider a function real-valued function with $C^{\infty}$-class on $\mathbb{R}$ such that:
(a) $f(0)=\frac{df}{dx}(0)=0,\ \frac{d^2f}{dx^2}(0)\neq 0.$
(b) For $x\neq 0,\ f(x)>0.$
Judge whether the following integrals $(i),\ (ii)$ converge or diverge, justify your answer.
$(i)$ \[\int\int_{|x_1|^2+|x_2|^2\leq 1} \frac{dx_1dx_2}{f(x_1)+f(x_2)}.\]
$(ii)$ \[\int\int_{|x_1|^2+|x_2|^2+|x_3|^2\leq 1} \frac{dx_1dx_2dx_3}{f(x_1)+f(x_2)+f(x_3)}.\]
[i]2010 Kyoto University, Master Course in Mathematics[/i]
In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$.
[asy]
Label f;
xaxis(0,60,blue);
yaxis(0,60,blue);
real f(real x)
{
return (x^2)/60;
}
draw(graph(f,0,53),red);
label("$y=x^2$",(30,15),E);
real f(real x)
{
return (x^2)/25;
}
draw(graph(f,0,38),red);
label("$y=2x^2$",(37,37^2/25),E);
real f(real x)
{
return (x^2)/10;
}
draw(graph(f,0,25),red);
label("$y=f(x)$",(24,576/10),W);
label("$O(0,0)$",(0,0),S);
dot((20,400/25));
dot((20,400/60));
label("$P$",(20,400/25),E);
label("$B$",(20,400/60),SE);
dot(((4000/25)^(0.5),400/25));
label("$A$",((4000/25)^(0.5),400/25),W);
draw((20,400/25)..((4000/25)^(0.5),400/25));
draw((20,400/25)..(20,400/60));
[/asy]
Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$.
created by kunny
Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.
Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$
Let $ f(x) \equal{} (1 \plus{} x)\cdot\sqrt{(2 \plus{} x^2)}\cdot\sqrt[3]{(3 \plus{} x^3)}$. Determine $ f'(1)$.
Find the minimum value of the following function $f(x) $ defined at $0<x<\frac{\pi}{2}$.
\[f(x)=\int_0^x \frac{d\theta}{\cos \theta}+\int_x^{\frac{\pi}{2}} \frac{d\theta}{\sin \theta}\]
Evaluate
\[\int_0^{\pi} a^x\cos bx\ dx,\ \int_0^{\pi} a^x\sin bx\ dx\ (a>0,\ a\neq 1,\ b\in{\mathbb{N^{+}}})\]
Own
Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.
Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$
Let $a_1=1$ and $a_n=n(a_{n-1}+1)$ for all $n\ge 2$ . Define :
$P_n=\left(1+\frac{1}{a_1}\right)...\left(1+\frac{1}{a_n}\right)$
Compute $\lim_{n\to \infty} P_n$
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that
\[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\]
Own
Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$.
(1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$.
(2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$.
(3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.
Evaluate the following definite integral.
\[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]
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.
Evaluate the following definite integral.
\[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]
Given mobile points $P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ on the $x$-$y$ plane.
Denote by $D$ the part in which line segment $PQ$ sweeps. Find the volume $V$ generated by a rotation of $D$ around the $x$-axis.
Find the indefinite integral $ \int \frac{x^3}{(x\minus{}1)^3(x\minus{}2)}\ dx$.