Found problems: 713
For a positive constant $ t$, let $ \alpha ,\ \beta$ be the roots of the quadratic equation $ x^2 \plus{} t^2x \minus{} 2t \equal{} 0$.
Find the minimum value of $ \int_{ \minus{} 1}^2 \left\{\left(x \plus{} \frac {1}{\alpha ^ 2}\right)\left(x \plus{} \frac {1}{\beta ^ 2}\right) \plus{} \frac {1}{\alpha \beta}\right\}dx.$
Define a function $f(x)=\int_0^{\frac{\pi}{2}} \frac{\cos |t-x|}{1+\sin |t-x|}dt$ for $0\leq x\leq \pi$.
Find the maximum and minimum value of $f(x)$ in $0\leq x\leq \pi$.
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 $\int_0^{\pi} \{f(x)\}^3dx.$
Prove that $ f(x)\equal{}\int_0^1 e^{\minus{}|t\minus{}x|}t(1\minus{}t)dt$ has maximal value at $ x\equal{}\frac 12$.
Find the volume of the solid defined by the inequality $ x^2 \plus{} y^2 \plus{} \ln (1 \plus{} z^2)\leq \ln 2$.
Note that you may not directively use double integral here for Japanese high school students who don't study it.
Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$.
(1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$.
(2) Find the value of $ nx_nx_{n \minus{} 1}$.
(3) Show that a sequence $ \{x_n\}$ is monotone decreasing.
(4) Find $ \lim_{n\to\infty} nx_n^2$.
Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$
Calculate the following indefinite integrals.
[1] $\int \frac{x}{\sqrt{5-x}}dx$
[2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$
[3] $\int (\sin x+\cos x)^2dx$
[4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$
[5]$\int (\sin x+\sin 2x)^2 dx$
Find $ \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}$.
Find continuous functions $ f(x),\ g(x)$ which takes positive value for any real number $ x$, satisfying $ g(x)\equal{}\int_0^x f(t)\ dt$ and $ \{f(x)\}^2\minus{}\{g(x)\}^2\equal{}1$.
Evaluate
\[\lim_{n\to\infty} n\int_0^\pi e^{-nx} \sin ^ 2 nx\ dx\]
Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$
Evaluate $\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.$
Let $ n\geq 2$ be natural number. Answer the following questions.
(1) Evaluate the definite integral $ \int_1^n x\ln x\ dx.$
(2) Prove the following inequality.
$ \frac 12n^2\ln n \minus{} \frac 14(n^2 \minus{} 1) < \sum_{k \equal{} 1}^n k\ln k < \frac 12n^2\ln n \minus{} \frac 14 (n^2 \minus{} 1) \plus{} n\ln n.$
(3) Find $ \lim_{n\to\infty} (1^1\cdot 2^2\cdot 3^3\cdots\cdots n^n)^{\frac {1}{n^2 \ln n}}.$
In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$.
[asy]
Label f;
xaxis(0,60,blue);
yaxis(0,60,blue);
real f(real x)
{
return (x^2)/60;
}
draw(graph(f,0,53),red);
label("$y=x^2$",(30,15),E);
real f(real x)
{
return (x^2)/25;
}
draw(graph(f,0,38),red);
label("$y=2x^2$",(37,37^2/25),E);
real f(real x)
{
return (x^2)/10;
}
draw(graph(f,0,25),red);
label("$y=f(x)$",(24,576/10),W);
label("$O(0,0)$",(0,0),S);
dot((20,400/25));
dot((20,400/60));
label("$P$",(20,400/25),E);
label("$B$",(20,400/60),SE);
dot(((4000/25)^(0.5),400/25));
label("$A$",((4000/25)^(0.5),400/25),W);
draw((20,400/25)..((4000/25)^(0.5),400/25));
draw((20,400/25)..(20,400/60));
[/asy]
Prove that :
\[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]
Let $a,\ b,\ c$ be positive real numbers such that $b^2>ac.$
Evaluate
\[\int_0^{\infty} \frac{dx}{ax^4+2bx^2+c}.\]
[i]1981 Tokyo University, Master Course[/i]
Given a parabola $C : y=1-x^2$ in $xy$-palne with the origin $O$. Take two points $P(p,\ 1-p^2),\ Q(q,\ 1-q^2)\ (p<q)$ on $C$.
(1) Express the area $S$ of the part enclosed by two segments $OP,\ OQ$ and the parabalola $C$ in terms of $p,\ q$.
(2) If $q=p+1$, then find the minimum value of $S$.
(3) If $pq=-1$, then find the minimum value of $S$.
Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$.
(1) Find the equations of the tangents $l_1,\ l_2$.
(2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.
For a positive constant number $ t$, let denote $ D$ the region surrounded by the curve $ y \equal{} e^{x}$, the line $ x \equal{} t$, the $ x$ axis and the $ y$ axis. Let $ V_x,\ V_y$ be the volumes of the solid obtained by rotating $ D$ about the $ x$ axis and the $ y$ axis respectively. Compare the size of $ V_x,\ V_y.$
A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions:
(i) $f_0(x)=e^{2x}+1$.
(ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$
Define sequences $\{a_n\},\ \{b_n\}$ by
\[a_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}d\theta,\ b_n=\int_{-\frac {\pi}6}^{\frac{\pi}6} e^{n\sin \theta}\cos \theta d\theta\ (n=1,\ 2,\ 3,\ \cdots).\]
(1) Find $b_n$.
(2) Prove that for each $n$, $b_n\leq a_n\leq \frac 2{\sqrt{3}}b_n.$
(3) Find $\lim_{n\to\infty} \frac 1{n}\ln (na_n).$
Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$
Evaluate
\[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]
(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that
\[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\]
Explain the fact by using graph. Note that you don't need to prove the statement.
(2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$,
Prove that there exists $ \theta$ such that
\[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]