Found problems: 713
Evaluate $ \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{1\plus{}\sin \theta \minus{}\cos \theta}\ d\theta$
Find the area of the region of the points such that the total of three tangent lines can be drawn to two parabolas $y=x-x^2,\ y=a(x-x^2)\ (a\geq 2)$ in such a way that there existed the points of tangency in the first quadrant.
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$
Evaluate $\int_0^1 f(t)dt.$
Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.
If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that
\[x_{0}^{2}<f(x_{0})<x_{0}^{3}\]
[Edit: $f$ is continuous]
Evaluate $\int_1^{e^2} \frac{(2x^2+2x+1)e^{x}}{\sqrt{x}}\ dx.$
$ \int_{\minus{}\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x\minus{}\sin x}\right|\ dx$.
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
Find the minimum value of $ \int_0^{\pi} \left(x \minus{} \pi a \minus{} \frac {b}{\pi}\cos x\right)^2dx$.
Let $\alpha$ be a solution satisfying the equation $|x|=e^{-x}.$ Let $I_n=\int_0^{\alpha} (xe^{-nx}+\alpha x^{n-1})dx\ (n=1,\ 2,\ \cdots).$
Find $\lim_{n\to\infty} n^2I_n.$
Evaluate
\[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]
Find the real number $a$ such that $\int_0^a \frac{e^x+e^{-x}}{2}dx=\frac{12}{5}.$
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.
For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$.
(1) Find the minimum value of $f(x)$.
(2) Evaluate $\int_0^1 f(x)\ dx$.
[i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]
Let $ a,\ b,\ c$ be real numbers such that $ a\geq b\geq c\geq 1$.
Prove the following inequality:
\[ \int_0^1 \{(1\minus{}ax)^3\plus{}(1\minus{}bx)^3\plus{}(1\minus{}cx)^3\minus{}3x\}\ dx\geq ab\plus{}bc\plus{}ca\minus{}\frac 32(a\plus{}b\plus{}c)\minus{}\frac 34abc.\]
Find $ \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.$
Let $(F_n)_{n\in{N^*}}$ be the Fibonacci sequence defined by
$F_1=1$, $F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for every $n\geq{2}$. Find
the limit: \[ \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}}) \]
Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.
Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$.
Find $\lim_{n\to\infty} f_n(x)$
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad \\
z\leq y^2 &\quad
\end{array}
\right.\]
Find the volume of the solid $V$.
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.
\[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]
In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis.
Note you may not use double integral.
Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{(2\sin \theta +1)\cos ^ 3 \theta}{(\sin ^ 2 \theta +1)^2}d\theta .$
[i]Proposed by kunny[/i]