Found problems: 713
Evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{2+\sin x}{1+\cos x}\ dx.$
Find $\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).$
Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure
which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis.
If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.
Find $\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}$.
[i]2011 Hosei University entrance exam/Design and Enginerring[/i]
Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.
Problem 3.
Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that:
(a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$
(b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$
For $ n \equal{} 1,\ 2,\ 3,\ \cdots$, let $ (p_n,\ q_n)\ (p_n > 0,\ q_n > 0)$ be the point of intersection of $ y \equal{} \ln (nx)$ and $ \left(x \minus{} \frac {1}{n}\right)^2 \plus{} y^2 \equal{} 1$.
(1) Show that $ 1 \minus{} q_n^2\leq \frac {(e \minus{} 1)^2}{n^2}$ to find $ \lim_{n\to\infty} q_n$.
(2) Find $ \lim_{n\to\infty} n\int_{\frac {1}{n}}^{p_n} \ln (nx)\ dx$.
Find $\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.$
(1) Let $f(x)$ be the differentiable and increasing function such that $f(0)=0.$Prove that $\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.$
(2) $g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)$ satisfies $\int_{-1}^{1}(px+q)g_{n}(x)dx=0$ for all linear equations $px+q.$
Find $a_{n},\ b_{n}.$
Prove the following inequality.
\[ \ln \frac {\sqrt {2009} \plus{} \sqrt {2010}}{\sqrt {2008} \plus{} \sqrt {2009}} < \int_{\sqrt {2008}}^{\sqrt {2009}} \frac {\sqrt {1 \minus{} e^{ \minus{} x^2}}}{x}\ dx < \sqrt {2009} \minus{} \sqrt {2008}\]
evaluate
\[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]
Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.
Evaluate
\[ \int_{\minus{}1}^1 \frac{1\plus{}2x^2\plus{}3x^4\plus{}4x^6\plus{}5x^8\plus{}6x^{10}\plus{}7x^{12}}{\sqrt{(1\plus{}x^2)(1\plus{}x^4)(1\plus{}x^6)}}dx.\]
Evaluate $ \int_0^1 \frac{x\minus{}e^{2x}}{x^2\minus{}e^{2x}}dx$.
Let $a>1$ be a constant. In the $xy$-plane, let $A(a,\ 0),\ B(a,\ \ln a)$ and $C$ be the intersection point of the curve $y=\ln x$ and the $x$-axis. Denote by $S_1$ the area of the part bounded by the $x$-axis, the segment $BA$ and the curve $y=\ln x$
(1) For $1\leq b\leq a$, let $D(b,\ \ln b)$. Find the value of $b$ such that the area of quadrilateral $ABDC$ is the closest to $S_1$ and find the area $S_2$.
(2) Find $\lim_{a\rightarrow \infty} \frac{S_2}{S_1}$.
[i]1992 Tokyo University entrance exam/Science[/i]
Find the volume of the solid obtained by revolving the region bounded by the graphs of $y=xe^{1-x}$ and $y=x$ around the $x$ axis.
Evaluate $ \int_1^e \frac {(1 \plus{} 2x^2)\ln x}{\sqrt {1 \plus{} x^2}}\ dx$.
Evaluate $ \int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx$
Last Edited, Sorry
kunny
Evaluate
\[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\]
Own
Let $f(x)$ be a polynomial. Prove that if $\int_0^1 f(x)g_n(x)\ dx=0\ (n=0,\ 1,\ 2,\ \cdots)$, then all coefficients of $f(x)$ are 0 for each case as follows.
(1) $g_n(x)=(1+x)^n$
(2) $g_n(x)=\sin n\pi x$
(3) $g_n(x)=e^{nx}$
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$,
$I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$
Find $\lim_{n\to\infty} I_n.$
[i]2009 Tokyo Institute of Technology, Master Course in Mathematics[/i]
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.
A function $f(x)$ satisfies $f(x)=f\left(\frac{c}{x}\right)$ for some real number $c(>1)$ and all positive number $x$.
If $\int_1^{\sqrt{c}} \frac{f(x)}{x} dx=3$, evaluate $\int_1^c \frac{f(x)}{x} dx$
Evaluate
\[\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin ^ 2 nx}{\sin x}dx-\sum_{k=1}^n \frac{1}{k}\right)\]
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.