Found problems: 713
Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$
Evaluate $ \int_0^\frac{\pi}{4} \frac{x\plus{}\sin x}{1\plus{}\cos x}\ dx$.
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.$
Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$
([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.
In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes.
1984 Tokyo University entrance exam/Science
Let $a$ be a positive constant. Find the value of $\ln a$ such that
\[\frac{\int_1^e \ln (ax)\ dx}{\int_1^e x\ dx}=\int_1^e \frac{\ln (ax)}{x}\ dx.\]
For a positive constant $a$, find the minimum value of $f(x)=\int_0^{\frac{\pi}{2}} |\sin t-ax\cos t|dt.$
Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$
Find the minimum value of $I(t)\ (t>1).$
In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.
Answer the following questions:
(1) Let $a$ be positive real number. Find $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$
(2) Evaluate $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$
35 points
Find the volume of the uion $ A\cup B\cup C$ of the three subsets $ A,\ B,\ C$ in $ xyz$ space such that:
\[ A\equal{}\{(x,\ y,\ z)\ |\ |x|\leq 1,\ y^2\plus{}z^2\leq 1\}\]
\[ B\equal{}\{(x,\ y,\ z)\ |\ |y|\leq 1,\ z^2\plus{}x^2\leq 1\}\]
\[ C\equal{}\{(x,\ y,\ z)\ |\ |z|\leq 1,\ x^2\plus{}y^2\leq 1\}\]
Find the non-constant function $f(x)$ such that $f(x)=x^2-\int_0^1 (f(t)+x)^2dt.$
Let $f(x)=\frac{e^x}{e^x+1}$
Prove the following equation.
\[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]
Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$
Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions.
(1) Express the coordiante of $ H$ in terms of $ t$.
(2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$.
(3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.
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
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {dx}{(1 \plus{} \cos x)^2}$.
Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]
Calculate: $\underset{n\to \infty }{\mathop{\lim }}\,\int_{0}^{1}{{{e}^{{{x}^{n}}}}dx}$
In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$.
Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.
For $x\geq 0$ find the value of $x$ by which $f(x)=\int_0^x 3^t(3^t-4)(x-t)dt$ is minimized.
Let $a$ be real number. Evaluate $\int_a^{a+\pi} |x|\cos x\ dx.$
Let $ a,\ b,\ c$ be postive constants.
Evaluate $ \int_0^1 \frac{2a\plus{}3bx\plus{}4cx^2}{2\sqrt{a\plus{}bx\plus{}cx^2}}\ dx$.
Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$.
(1) Find the maximum and minimum values of $ f(\theta)$.
(2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.
Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized.
Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$
[i]2010 Saitama University entrance exam/Mathematics[/i]
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