Found problems: 713
Let $a_n=\int_0^{\frac{\pi}{2}} \sin 2t\ (1-\sin t)^{\frac{n-1}{2}}dt\ (n=1,2,\cdots)$
Evaluate
\[\sum_{n=1}^{\infty} (n+1)(a_n-a_{n+1})\]
Define the sequence $\{x_{n}\}$ by $x_{0}=a\in (0,1)$ and $x_{n+1}=\frac{4}{\pi^{2}}(\cos^{-1}x_{n}+\frac{\pi}{2})\sin^{-1}x_{n}(n=0,1,2,...)$. Show that the sequence converges and find its limit.
Calculate the following indefinite integrals.
[1] $\int (x^2+4x-3)^2(x+2)dx$
[2] $\int \frac{\ln x}{x(\ln x+1)}dx$
[3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$
[4] $\int \frac{dx}{\sin x\cos ^ 2 x}$
[5] $\int \sqrt{1-3x}\ dx$
Prove the following inequality.
\[ \frac{\pi}{4}\sqrt{\frac{3}{2}\plus{}\sqrt{2}}<\int_0^{\frac{\pi}{2}} \sqrt{1\minus{}\frac 12\sin ^ 2 x}\ dx<\frac{\sqrt{3}}{4}\pi\]
Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.
Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$
[i]2010 Kumamoto University entrance exam/Medicine[/i]
Let $l$ be the tangent line at the point $(t,\ t^2)\ (0<t<1)$ on the parabola $C: y=x^2$. Denote by $S_1$ the area of the part enclosed by $C,\ l$ and the $x$-axis, denote by $S_2$ of the area of the part enclosed by $C,\ l$ and the line $x=1$. Find the minimum value of $S_1+S_2$.
Calculate the following indefinite integrals.
(1) $ \int \frac {3x \plus{} 4}{x^2 \plus{} 3x \plus{} 2}dx$
(2) $ \int \sin 2x\cos 2x\cos 4x\ dx$
(3) $ \int xe^{x}dx$
(4) $ \int 5^{x}dx$
Calculate the following indefinite integrals.
[1] $\int \cos \left(2x-\frac{\pi}{3}\right)dx$
[2]$\int \frac{dx}{\cos ^2 (3x+4)}$
[3]$\int (x-1)\sqrt[3]{x-2}dx$
[4]$\int x\cdot 3^{x^2+1}dx$
[5]$\int \frac{dx}{\sqrt{1-x}}dx$
Prove that for positive number $t$, the function $F(t)=\int_0^t \frac{\sin x}{1+x^2}dx$ always takes positive number.
1972 Tokyo University of Education entrance exam
If the normal to the curve $x^{\frac{2}{3}}+y^{\frac23}=a^{\frac23}$ at some point makes an angle $\theta$ with the $X$-axis, show that the equation of the normal is
\[y\cos\theta-x\sin\theta=a\cos 2\theta\]
Evaluate $ \int_0^{\ln 2} (x\minus{}\ln 2)e^{\minus{}2\ln (1\plus{}e^x)\plus{}x\plus{}\ln 2}dx$.
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$
Evaluate
\[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
Evaluate
\[\sum_{k=0}^{2004} \int_{-1}^1 \frac{\sqrt{1-x^2}}{\sqrt{k+1}-x}dx\]
Evaluate $ \sum_{k\equal{}1}^n \int_0^{\pi} (\sin x\minus{}\cos kx)^2dx.$
Find the continuous function $f(x)$ such that the following equation holds for any real number $x$.
\[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\]
[i]1977 Keio University entrance exam/Medicine[/i]
Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous.
Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.$
Find the minimum value of $\int_0^1 \frac{|t-x|}{t+1}dt$
Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$
Evaluate
\[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]
Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 points
Evaluate the following integrals.
(1) $\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.$
(2) $\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.$