Found problems: 713
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
For real numbers $a,\ b$ with $0\leq a\leq \pi,\ a<b$, let $I(a,\ b)=\int_{a}^{b} e^{-x} \sin x\ dx.$
Determine the value of $a$ such that $\lim_{b\rightarrow \infty} I(a,\ b)=0.$
Let $ K$ be the figure bounded by the curve $ y\equal{}e^x$ and 3 lines $ x\equal{}0,\ x\equal{}1,\ y\equal{}0$ in the $ xy$ plane.
(1) Find the volume of the solid formed by revolving $ K$ about the $ x$ axis.
(2) Find the volume of the solid formed by revolving $ K$ about the $ y$ axis.
Evaluate the following integrals.
(1) Meiji University
$\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.$
(2) Tokyo University of Science
$\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.$
Consider two curves $y=\sin x,\ y=\sin 2x$ in $0\leq x\leq 2\pi$.
(1) Let $(\alpha ,\ \beta)\ (0<\alpha <\pi)$ be the intersection point of the curves. If $\sin x-\sin 2x$ has a local minimum at $x=x_1$ and a local maximum at $x=x_2$, then find the values of $\cos x_1,\ \cos x_1\cos x_2$.
(2) Find the area enclosed by the curves, then find the volume of the part generated by a rotation of the part of $\alpha \leq x\leq \pi$ for the figure about the line $y=-1$.
[i]2011 Kyorin University entrance exam/Medicine [/i]
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: $\lim_{n\to\infty} \frac{1}{2n} \ln\binom{2n}{n}$
Let $P_k$ be a point whose $x$-coordinate is $1+\frac{k}{n}\ (k=1,\ 2,\ \cdots,\ n)$ on the curve $y=\ln x$. For $A(1,\ 0)$, find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^{n} \overline{AP_k}^2.$
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.$
Evaluate $ \int_0^{\frac{\pi}{4}} \{(x\sqrt{\sin x}\plus{}2\sqrt{\cos x})\sqrt{\tan x}\plus{}(x\sqrt{\cos x}\minus{}2\sqrt{\sin x})\sqrt{\cot x}\}\ dx.$
(1) Evaluate $ \int_1^{3\sqrt{3}} \left(\frac{1}{\sqrt[3]{x^2}}\minus{}\frac{1}{1\plus{}\sqrt[3]{x^2}}\right)\ dx.$
(2) Find the positive real numbers $ a,\ b$ such that for $ t>1,$ $ \lim_{t\rightarrow \infty} \left(\int_1^t \frac{1}{1\plus{}\sqrt[3]{x^2}}\ dx\minus{}at^b\right)$ converges.
Find $\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}$.
For $a>1$, evaluate $\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.$
Evaluate $ \frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}$
Calculate the following indefinite integrals.
[1] $\int \ln (x^2-1)dx$
[2] $\int \frac{1}{e^x+1}dx$
[3] $\int (ax^2+bx+c)e^{mx}dx\ (abcm\neq 0)$
[4] $\int \left(\tan x+\frac{1}{\tan x}\right)^2 dx$
[5] $\int \sqrt{1-\sin x}dx$
Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$
Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$
(1) Find $f(0).$
(2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$
(3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$
(4) Find $\lim_{x\rightarrow +\infty} g(x)$
Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.
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.
A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$
Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$
Let $C$ be the point of intersection of the tangent lines $l,\ m$ at $A(a,\ a^2),\ B(b,\ b^2)\ (a<b)$ on the parabola $y=x^2$ respectively.
When $C$ moves on the parabola $y=\frac 12 x^2-x-2$, find the minimum area bounded by 2 lines $l,\ m$ and the parabola $y=x^2$.
([b]4[/b]) Let $ f(x) \equal{} \sin^6\left(\frac {x}{4}\right) \plus{} \cos^6\left(\frac {x}{4}\right)$ for all real numbers $ x$. Determine $ f^{(2008)}(0)$ (i.e., $ f$ differentiated $ 2008$ times and then evaluated at $ x \equal{} 0$).
Find the minimum value of $\int_0^{\frac{\pi}{2}} |a\sin 2x-\cos ^ 2 x|dx\ (a>0).$
2009 Shimane University entrance exam/Medicine
Find $\lim_{n\to\infty} \int_0^{\pi} x|\sin 2nx| dx\ (n=1,\ 2,\ \cdots)$.
[i]1992 Japan Women's University entrance exam/Physics, Mathematics[/i]
Use $ \frac{d}{dx} \ln (2x\plus{}\sqrt{4x^2\plus{}1}),\ \frac{d}{dx}(x\sqrt{4x^2\plus{}1})$ to evaluate $ \int_0^1 \sqrt{4x^2\plus{}1}dx$.
Evaluate
\[\int_0^{\pi} \{(1-x\sin 2x)e^{\cos ^2 x}+(1+x\sin 2x)e^{\sin ^ 2 x}\}\ dx.\]
Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.