Found problems: 713
Find the maximum and minimum values of $ \int_0^{\pi} (a\sin x \plus{} b\cos x)^3dx$ for $ |a|\leq 1,\ |b|\leq 1$.
Note that you are not allowed to solve in using partial differentiation here.
Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$
Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.
Evaluate
\[\int_{-\pi}^{\pi} (\cos2x)(\cos 2^2x)\cdots (\cos 2^{2006}x)dx\]
For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.
For $f(x)=\frac{1}{x}\ (x>0)$, prove the following inequality.
\[f\left(t+\frac 12 \right)\leq \int_t^{t+1} f(x)\ dx\leq \frac 16\left\{f(t)+4f\left(t+\frac 12\right)+f(t+1)\right\}\]
Find $\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).$
Evaluate $\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.$
created by kunny
For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows.
Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx.
(1) Find $f(\sqrt{3})$
(2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$
(3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.
Let $p$ be a constant number such that $0<p<1$.
Evaluate
\[\sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\displaystyle \int_0^1 x^k (1-x)^{2004-k} dx}\]
Answer the following questions:
(1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$.
Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$.
(2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$
Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.
Evaluate $\int_1^3 \frac{\ln (x+1)}{x^2}dx$.
[i]2010 Hirosaki University entrance exam[/i]
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 $\frac{S(t)}{\sqrt{t}}.$
Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$
Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$
Find the minimum value of $a\ (0<a<1)$ for which the following definite integral is minimized.
\[ \int_0^{\pi} |\sin x-ax|\ dx \]
Consider a parameterized curve $C: x=e^{-t}\cos t,\ y=e^{-t}\sin t\left (0\leq t\leq \frac{\pi}{2}\right).$
(1) Find the length $L$ of $C$.
(2) Find the area $S$ of the region enclosed by the $x,\ y$ axis and $C$.
Please solve the problem without using the formula of area for polar coordinate for Japanese High School Students who don't study it in High School.
[i]1997 Kyoto University entrance exam/Science[/i]
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.
Let $C: y=\ln x$. For each positive integer $n$, denote by $A_n$ the area of the part enclosed by the line passing through two points $(n,\ \ln n),\ (n+1,\ \ln (n+1))$ and denote by $B_n$ that of the part enclosed by the tangent line at the point $(n,\ \ln n)$, $C$ and the line $x=n+1$. Let $g(x)=\ln (x+1)-\ln x$.
(1) Express $A_n,\ B_n$ in terms of $n,\ g(n)$ respectively.
(2) Find $\lim_{n\to\infty} n\{1-ng(n)\}$.
Let $P(x,\ y)\ (x>0,\ y>0)$ be a point on the curve $C: x^2-y^2=1$. If $x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)$, then find the area bounded by the line $OP$, the $x$ axis and the curve $C$ in terms of $u$.
Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.
The function $f_n (x)\ (n=1,2,\cdots)$ is defined as follows.
\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]
Evaluate
\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]
Show that there exists the maximum value of the function $f(x,\ y)=(3xy+1)e^{-(x^2+y^2)}$ on $\mathbb{R}^2$, then find the value.
Let $0<a<b$.Prove the following inequaliy.
\[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]
Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]