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.

Evaluate $ \int_0^{2(2\plus{}\sqrt{3})} \frac{16}{(x^2\plus{}4)^2}\ dx$.

For a constant $c$, a sequence $a_n$ is defined by $a_n=\int_c^1 nx^{n-1}\left(\ln \left(\frac{1}{x}\right)\right)^n dx\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\lim_{n\to\infty} a_n$.

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.

Find the minimum value of $\int_{-1}^1 \sqrt{|t-x|}\ dt$

Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

On the $x$-$y$ plane, 3 half-lines $y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)$ intersect with the circle with the center the origin $O$, radius $r\geq 1$ at $A,\ B,\ C$ respectively. Note that $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$. If the area of quadrilateral $OABC$ is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .$ [i]2011 Waseda University of Education entrance exam/Science[/i]

Let $ T \equal{} \int_0^{\ln2} \frac {2e^{3x} \plus{} e^{2x} \minus{} 1} {e^{3x} \plus{} e^{2x} \minus{} e^x \plus{} 1}dx$. Evaluate $ e^T$.

In the $xy$-plane, for $a>1$ denote by $S(a)$ the area of the figure bounded by the curve $y=(a-x)\ln x$ and the $x$-axis. Find the value of integer $n$ for which $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$ is non-zero real number.

Let $a,\ b$ be positive real numbers with $a<b$. Define the definite integrals $I_1,\ I_2,\ I_3$ by $I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx$. (1) Find the value of $I_1+\frac 12I_2$ in terms of $a,\ b$. (2) Find the value of $I_2-\frac 32I_3$ in terms of $a,\ b$. (3) For a positive integer $n$, define $K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx$. Find the value of $\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n$. [i]2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring[/i]

Find the limit \[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]

Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$. created by kunny

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]

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$.

Find $ \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx$.

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

Find the minimum value of $\int_0^1 \{\sqrt{x}-(a+bx)\}^2dx$. Please solve the problem without using partial differentiation for those who don't learn it. 1961 Waseda University entrance exam/Science and Technology

For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$. (1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$. (2) Find $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$.

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

Compare the size of the definite integrals? \[ \int_0^{\frac {\pi}{4}} x^{2008}\tan ^{2008}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2009}\tan ^{2009}x\ dx,\ \int_0^{\frac {\pi}{4}} x^{2010}\tan ^{2010}x\ dx\]

Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the parabola by $S(t).$ Find the minimum value of $\frac{S(t)}{\sqrt{t}}.$

$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ 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) $\frac{\pi}{3}-\frac 16\leq \int_0^{\frac{\sqrt{3}}{2}} \frac{1}{\sqrt{1-x^4}}dx\leq \frac{\pi}{3}.$

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 $\lim_{n\to\infty} c_n.$