Found problems: 713
Consider a curve $C$ on the $x$-$y$ plane expressed by $x=\tan \theta ,\ y=\frac{1}{\cos \theta}\left (0\leq \theta <\frac{\pi}{2}\right)$.
For a constant $t>0$, let the line $l$ pass through the point $P(t,\ 0)$ and is perpendicular to the $x$-axis,intersects with the curve $C$ at $Q$. Denote by $S_1$ the area of the figure bounded by the curve $C$, the $x$-axis, the $y$-axis and the line $l$, and denote by $S_2$ the area of $\triangle{OPQ}$. Find $\lim_{t\to\infty} \frac{S_1-S_2}{\ln t}.$
A sequence of polynomial $f_n(x)\ (n=0,1,2,\cdots)$ satisfies $f_0(x)=2,f_1(x)=x$,
\[f_n(x)=xf_{n-1}(x)-f_{n-2}(x),\ (n=2,3,4,\cdots)\]
Let $x_n\ (n\geqq 2)$ be the maximum real root of the equation $f_n(x)=0\ (|x|\leqq 2)$
Evaluate
\[\lim_{n\to\infty} n^2 \int_{x_n}^2 f_n(x)dx\]
Answer the following questions:
(1) Find the value of $a$ for which $S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx$ is minimized, then find the minimum value.
(2) Find the vlues of $p,\ q$ for which $T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx$ is minimized, then find the minimum value.
For every natural number $n$ let:
$a_n=ln(1+2e+4e^4+\dots+2ne^{n^2})$. Find:
\[ \displaystyle{\lim_{n \to \infty}\frac{a_n}{n^2}} \].
(1) Evaluate the following definite integrals.
(a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$
(b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$
(c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$
(2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that
\[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\]
, then find the volume of the solid.
[i]1984 Yamanashi Medical University entrance exam[/i]
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$
Evaluate $ \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx$.
Evaluate $\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.$
Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.
Find the value of $ a$ for which the area of the figure surrounded by $ y \equal{} e^{ \minus{} x}$ and $ y \equal{} ax \plus{} 3\ (a < 0)$ is minimized.
A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$.
Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.
Prove that :
\[\int_1^{\sqrt{e}} (\ln x)^n dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^m\ (n=1,\ 2,\ \cdots)\]
[i]2010 Miyazaki University entrance exam/Medicine[/i]
For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$
Answer the following questions.
(1) By setting $ x\plus{}\sqrt{x^2\minus{}1}\equal{}t$, find the indefinite integral $ \int \sqrt{x^2\minus{}1}\ dx$.
(2) Given two points $ P(p,\ q)\ (p>1,\ q>0)$ and $ A(1,\ 0)$ on the curve $ x^2\minus{}y^2\equal{}1$. Find the area $ S$ of the figure bounded by two lines $ OA,\ OP$ and the curve in terms of $ p$.
(3) Let $ S\equal{}\frac{\theta}{2}$. Express $ p,\ q$ in terms of $ \theta$.
Prove the following inequality for any integer $ N\geq 4$.
\[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]
Calculate the following indefinite integrals.
[1] $\int \sqrt{x}(\sqrt{x}+1)^2 dx$
[2] $\int (e^x+2e^{x+1}-3e^{x+2})dx$
[3] $\int (\sin ^2 x+\cos x)\sin x dx$
[4] $\int x\sqrt{2-x} dx$
[5] $\int x\ln x dx$
Let $f(x)$ be a function defined in $0<x<\frac{\pi}{2}$ satisfying:
(i) $f\left(\frac{\pi}{6}\right)=0$
(ii) $f'(x)\tan x=\int_{\frac{\pi}{6}}^x \frac{2\cos t}{\sin t}dt$.
Find $f(x)$.
[i]1987 Sapporo Medical University entrance exam[/i]
Evaluate
\[\int_0^1 (1-x^2)^n dx\ (n=0,1,2,\cdots)\]
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$
Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.
Evaluate the following definite integral.
\[\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx\]
Prove the following inequality.
\[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
For a function satisfying $ f'(x) > 0$ for $ a\leq x\leq b$, let $ F(x) \equal{} \int_a^b |f(t) \minus{} f(x)|\ dt$. For what value of $ x$ is $ F(x)$ is minimized?
Evaluate $ \int_0^{\frac {\pi}{2}} \frac {(\sin x)^{\cos x}}{(\cos x)^{\sin x} \plus{} (\sin x)^{\cos x}} dx$.