The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.

For $0\leq a\leq 2$, find the minimum value of $\int_0^2 \left|\frac{1}{1+e^x}-\frac{1}{1+e^a}\right|\ dx.$ [i]2010 Kyoto Institute of Technology entrance exam/Textile e.t.c.[/i]

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

Let $a,\ b$ be positive constants. Evaluate \[\int_0^1 \frac{\ln \frac{(x+a)^{x+a}}{(x+b)^{x+b}}}{(x+a)(x+b)\ln (x+a)\ln (x+b)}\ dx.\]

Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$ Find the minimum value of $I(t)\ (t>1).$

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

(1) For integer $n=0,\ 1,\ 2,\ \cdots$ and positive number $a_{n},$ let $f_{n}(x)=a_{n}(x-n)(n+1-x).$ Find $a_{n}$ such that the curve $y=f_{n}(x)$ touches to the curve $y=e^{-x}.$ (2) For $f_{n}(x)$ defined in (1), denote the area of the figure bounded by $y=f_{0}(x), y=e^{-x}$ and the $y$-axis by $S_{0},$ for $n\geq 1,$ the area of the figure bounded by $y=f_{n-1}(x),\ y=f_{n}(x)$ and $y=e^{-x}$ by $S_{n}.$ Find $\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).$

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.

Evaluate $\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.$

Prove that : \[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\] [i]2010 Miyazaki University entrance exam/Medicine[/i]

Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.

Let $f(x)$ be a differentiable function. Find the value of $x$ for which \[\{f(x)\}^2+(e+1)f(x)+1+e^2-2\int_0^x f(t)dt-2f(x)\int_0^x f(t)dt+2\left\{\int_0^x f(t)dt\right\}^2\] is minimized. [i]1978 Tokyo Medical College entrance exam[/i]

In the $xyz$ space with the origin $O$, Let $K_1$ be the surface and inner part of the sphere centered on the point $(1,\ 0,\ 0)$ with radius 2 and let $K_2$ be the surface and inner part of the sphere centered on the point $(-1,\ 0,\ 0)$ with radius 2. For three points $P,\ Q,\ R$ in the space, consider points $X,\ Y$ defined by \[\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).\] (1) When $P,\ Q$ move every cranny in $K_1,\ K_2$ respectively, find the volume of the solid generated by the whole points of the point $X$. (2) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1$. (3) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1\cup K_2$.

Determine the pair of constant numbers $ a,\ b,\ c$ such that for a quadratic function $ f(x) \equal{} x^2 \plus{} ax \plus{} b$, the following equation is identity with respect to $ x$. \[ f(x \plus{} 1) \equal{} c\int_0^1 (3x^2 \plus{} 4xt)f'(t)dt\] .

Prove that exists sequences $ (a_n)_{n\ge 0}$ with $ a_n\in \{\minus{}1,\plus{}1\}$, for any $ n\in \mathbb{N}$, such that: \[ \lim_{n\rightarrow \infty}\left(\sqrt{n\plus{}a_1}\plus{}\sqrt{n\plus{}a_2}\plus{}...\plus{}\sqrt{n\plus{}a_n}\minus{}n\sqrt{n\plus{}a_0}\right)\equal{}\frac{1}{2}\]

Evaluate \[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]

A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$. Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$. [i]1997 Hokkaido University entrance exam/Science[/i]

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:

Prove the following inequality. \[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]

Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$.　Find the minimum area of the part bounded by the curve $ C$ and two lines $ l,\ m$.

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

Find the limit $\lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n k\ln \left(\frac{n^2+(k-1)^2}{n^2+k^2}\right).$ [i]2010 Yokohama National University entrance exam/Engineering, 2nd exam[/i]

(1) 2009 Kansai University entrance exam Calculate $ \int \frac{e^{\minus{}2x}}{1\plus{}e^{\minus{}x}}\ dx$. (2) 2009 Rikkyo University entrance exam/Science Evaluate $ \int_0^ 1 \frac{2x^3}{1\plus{}x^2}\ dx$.