Found problems: 713
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
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$
Find $\lim_{n\to\infty} f_n(1).$
Evaluate $\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.$
Evaluate $ \int_0^2 \frac{1}{\sqrt {1 \plus{} x^3}}\ dx$.
Prove that :
\[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]
Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$.
Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis.
[i]2011 Yokohama National Universty entrance exam/Engineering[/i]
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 $e^3>20.$
(1) For $ 0 < x < 1$, prove that $ (\sqrt {2} \minus{} 1)x \plus{} 1 < \sqrt {x \plus{} 1} < \sqrt {2}.$
(2) Find $ \lim_{a\rightarrow 1 \minus{} 0} \frac {\int_a^1 x\sqrt {1 \minus{} x^2}\ dx}{(1 \minus{} a)^{\frac 32}}$.
In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.
[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$
[2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$
[3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$
[4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$
[5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$
Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$
Let $a,\ b$ real numbers such that $a>1,\ b>1.$
Prove the following inequality.
\[\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2\]
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$
Find the value of $\frac{I_1}{I_2}.$
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
Evaluate the following definite integrals:
(a) $\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx$
(b) $\int_0^{\frac 13} xe^{3x}\ dx$
(c) $\int_e^{e^e} \frac{1}{x\ln x}\ dx$
(d) $\int_2^3 \frac{x^2+1}{x(x+1)}\ dx$
When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$
Evaluate
\[\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx\]
Evaluate
\[\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}\]
For $ x\geq 0,$ define a function $ f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.
Evaluate $ \int_0^{100\pi } f(x)\ dx.$
Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $x$-axis and denote $T$ by the area of the figure bounded by the curves $y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$ and the $x$-axis. Find the value of $a$ such that $ S: T=3: 1$.
(1) Show the following inequality for every natural number $ k$.
\[ \frac {1}{2(k \plus{} 1)} < \int_0^1 \frac {1 \minus{} x}{k \plus{} x}dx < \frac {1}{2k}\]
(2) Show the following inequality for every natural number $ m,\ n$ such that $ m > n$.
\[ \frac {m \minus{} n}{2(m \plus{} 1)(n \plus{} 1)} < \log \frac {m}{n} \minus{} \sum_{k \equal{} n \plus{} 1}^{m} \frac {1}{k} < \frac {m \minus{} n}{2mn}\]
A cubic funtion $ y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0)$ intersects with the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\ \gamma$.
Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$
Let $r$ be a positive integer. Determine the value of $a$ for which the limit value $\lim_{n\to\infty} \frac{\sum_{k=1}^n k^r}{n^a} $ has a non zero finite value, then find the limit value.
1956 Tokyo Institute of Technology entrance exam
Let $m,\ n$ be positive integer such that $2\leq m<n$.
(1) Prove the inequality as follows.
\[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\]
(2) Prove the inequality as follows.
\[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\]
(3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows.
\[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]