Found problems: 713
(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$.
(2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.
Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.
Evaluate $\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx$.
Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n$.
Find $ \lim_{n\to\infty}(a_{n}\minus{}\ln n)$.
Calculate the following indefinite integrals.
[1] $\int \frac{dx}{1+\cos x}$
[2] $\int x\sqrt{x^2-1}dx$
[3] $\int a^{-\frac{x}{2}}dx\ \ (a>0,a\neq 1)$
[4] $\int \frac{\sin ^ 3 x}{1+\cos x}dx$
[5] $\int e^{4x}\sin 2x dx$
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad \\
z\leq y^2 &\quad
\end{array}
\right.\]
Find the volume of the solid $V$.
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $S(x)$ at the hight $x$.
(2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$
Evaluate $ \int_0^1 \frac{x}{\{(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\plus{}(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.
Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$,
$ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\
y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$
When the point $ P$ moves on the curve $ C$,
(1) Find the magnitude of acceleralation of the point $ P$ at time $ t$.
(2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$.
(3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis.
Edited.
Given are the curve $y=x^2+x-2$ and a curve which is obtained by tranfering the curve symmetric with respect to the point $(p,\ 2p)$. Let $p$ change in such a way that these two curves intersects, find the maximum area of the part bounded by these curves.
[i]1978 Nagasaki University entrance exam/Economics[/i]
Evaluate the integrals as follows.
(1) $\int \frac{x^2}{2-x}\ dx$
(2) $\int \sqrt[3]{x^5+x^3}\ dx$
(3) $\int_0^1 (1-x)\cos \pi x\ dx$
Evaluate $ \int_0^{2008} \left(3x^2 \minus{} 8028x \plus{} 2007^2 \plus{} \frac {1}{2008}\right)\ dx$.
Evaluate
\[\int_{\frac{\pi}{2}}^{\frac{2\pi}{3}} \frac{1}{\sin x \sqrt{1-\cos x}}dx\]
Find the continuous function such that $ f(x)\equal{}\frac{e^{2x}}{2(e\minus{}1)}\int_0^1 e^{\minus{}y}f(y)dy\plus{}\int_0^{\frac 12} f(y)dy\plus{}\int_0^{\frac 12}\sin ^ 2(\pi y)dy$.
Let $f:\mathbb{R} \to \mathbb{R}$ be a function that is a function that is differentiable $n+1$ times for some positive integer $n$ . The $i^{th}$ derivative of $f$ is denoted by $f^{(i)}$ . Suppose-
$f(1)=f(0)=f^{(1)}(0)=...=f^{(n)}(0)=0$.
Prove that $f^{(n+1)}(x)=0$ for some $x \in (0,1)$
For a function $ f(x)\equal{}\int_x^{\frac{\pi}{4}\minus{}x} \log_4 (1\plus{}\tan t)dt\ \left(0\leq x\leq \frac{\pi}{8}\right)$, answer the following questions.
(1) Find $ f'(x)$.
(2) Find the $ n$ th term of the sequence $ a_n$ such that $ a_1\equal{}f(0),\ a_{n\plus{}1}\equal{}f(a_n)\ (n\equal{}1,\ 2,\ 3,\ \cdots)$.
Find the area of the figure bounded by two curves $y=x^4,\ y=x^2+2$.
A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.
$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$
Evaluate
\[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]
Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral.
\[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]
For any quadratic functions $ f(x)$ such that $ f'(2)\equal{}1$, evaluate $ \int_{2\minus{}\pi}^{2\plus{}\pi}f(x)\sin\left(\frac{x}{2}\minus{}1\right) dx$.
Evaluate
\[{{\int_{e^{e^{e}}}^{e^{e^{e^{e}}}}} \frac{dx}{x\ln x\cdot \ln (\ln x)\cdot \ln \{\ln (\ln x)\}}}\]
When the parabola which has the axis parallel to $y$ -axis and passes through the origin touch to the rectangular hyperbola $xy=1$ in the first quadrant moves,
prove that the area of the figure sorrounded by the parabola and the $x$-axis is constant.
Evaluate $\int_0^{\infty} \frac{\ln (1+e^{4x})}{e^x}dx.$