Found problems: 713
A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$.
Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$.
[i]1997 Hokkaido University entrance exam/Science[/i]
Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by
\[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\]
(1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$
(2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$
Find the area of the part bounded by two curves $y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $x$-axis.
1956 Tokyo Institute of Technology entrance exam
Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers.
Prove the following equations.
(1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$
(2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$
(3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$
Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$
Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.
Calculate $\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx$
Let $ \sigma \in S_n$ and $ \alpha <2$. Evaluate$ \displaystyle\lim_{n\to\infty} \displaystyle\sum_{k\equal{}1}^{n}\frac{\sigma (k)}{k^{\alpha}}$.
Let $n$ be a positive integer.
(1) For a positive integer $k$ such that $1\leq k\leq n$, Show that :
\[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\]
(2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$
If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$
(3) Find $\lim_{n\to\infty} S_n.$
For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$.
Find the maximum area of $S(a)$.
Evaluate $ \int_1^e \frac{\sqrt[3]{x}}{x(\sqrt{x}\plus{}\sqrt[3]{x})}\ dx$.
Suppose a function $f(x)$ is continuous on $[-1,\ 1]$ and satisfies the condition :
1) $f(-1)\geq f(1).$
2) $x+f(x)$ is non decreasing function.
3) $\int_{-1}^ 1 f(x)\ dx=0.$
Show that $\int_{-1}^1 f(x)^2dx\leq \frac 23.$
Let $m,\ n$ be non negative integers.
Calculate
\[\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}. \]
where $_{i}C_{j}$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
Let $ a,\ b$ are postive constant numbers.
(1) Differentiate $ \ln (x\plus{}\sqrt{x^2\plus{}a})\ (x>0).$
(2) For $ a\equal{}\frac{4b^2}{(e\minus{}e^{\minus{}1})^2}$, evaluate $ \int_0^b \frac{1}{\sqrt{x^2\plus{}a}}\ dx.$
Let $n$ be a positive integer.
(1) For a positive integer $k$ such that $1\leq k\leq n$, Show that :
\[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\]
(2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$
If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$
(3) Find $\lim_{n\to\infty} S_n.$
Define two functions $f(t)=\frac 12\left(t+\frac{1}{t}\right),\ g(t)=t^2-2\ln t$. When real number $t$ moves in the range of $t>0$, denote by $C$ the curve by which the point $(f(t),\ g(t))$ draws on the $xy$-plane.
Let $a>1$, find the area of the part bounded by the line $x=\frac 12\left(a+\frac{1}{a}\right)$ and the curve $C$.
For a function $f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1)$, answer the following questions:
(1) Find $f'(x)$.
(2) Sketch the graph of $y=f(x)$.
(3) Let $P$ be a mobile point on the curve $y=f(x)$ and $Q$ be a point which is on the tangent at $P$ on the curve $y=f(x)$ and such that $PQ=1$. Note that the $x$-coordinate of $Q$ is les than that of $P$. Find the locus of $Q$.
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.
For a non negative integer $n$, set t $I_n=\int_0^{\frac{\pi}{4}} \tan ^ n x\ dx$ to answer the following questions:
(1) Calculate $I_{n+2}+I_n.$
(2) Evaluate the values of $I_1,\ I_2$ and $I_3.$
1978 Niigata university entrance exam
Evaluate $\int_{-\frac{\pi}{3}}^{\frac{\pi}{6}} \left|\frac{4\sin x}{\sqrt{3}\cos x-\sin x}\right|dx.$
2009 Kumamoto University entrance exam/Medicine
Let $a_n$ be the area of the part enclosed by the curve $y=x^n\ (n\geq 1)$, the line $x=\frac 12$ and the $x$ axis.
Prove that :
\[0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}\]
Show that a function $ f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt$ is continuous at $ x\equal{}0$.
Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$.
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Let $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$ for real number $ p > 0$. Answer the following questions.
(1) Find the value of $ x \equal{} s_p$ for which $ f(x)$ is minimal and draw the graph of $ y \equal{} f(x)$.
(2) Let $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$. Find the value of $ t \equal{} t_p$ for which $ g(t)$ is minimal.
(3) Use the fact $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$ to find the limit $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$.
Let $f(x)=\lim_{n\to\infty} \frac{\sin^{n+2}x+\cos^{n+2}x}{\sin^n x+\cos^n x}$ for $0\leq x\leq \frac{\pi}2.$
Evaluate $\int_0^{\frac{\pi}2} f(x)\ dx.$