Consider a parameterized curve $ C: x \equal{} e^{ \minus{} t}\cos t,\ y \equal{} e^{ \minus{} t}\sin t\ \left(0\leq t\leq \frac {\pi}{2}\right).$ (1) Find the length $ L$ of $ C$. (2) Find the area $ S$ of the region bounded by $ C$, the $ x$ axis and $ y$ axis. You may not use the formula $ \boxed{\int_a^b \frac {1}{2}r(\theta)^2d\theta }$ here.

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

Let $T$ be a period of a function $f(x)=|\cos x|\sin x\ (-\infty,\ \infty).$ Find $\lim_{n\to\infty} \int_0^{nT} e^{-x}f(x)\ dx.$

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

Prove the following inequality. \[ \frac{\sqrt{2}}{4}\minus{}\frac 12\minus{}\frac 14\ln 2<\int_0^{\frac{\pi}{4}} \ln \cos x\ dx<\frac 38\pi\plus{}\frac 12\minus{}\ln \ (3\plus{}2\sqrt{2})\]

Calculate the following indefinite integrals. [1] $\int \frac{dx}{e^x-e^{-x}}$ [2] $\int e^{-ax}\cos 2x dx\ (a\neq 0)$ [3] $\int (3^x-2)^2 dx$ [4] $\int \frac{x^4+2x^3+3x^2+1}{(x+2)^5}dx$ [5] $\int \frac{dx}{1-\cos x}dx$

Suppose that $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$. Find all possible values for which $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$ can be valued.

Let $f(x)=\int_0^ x \frac{1}{1+t^2}dt.$ For $-1\leq x<1$, find $\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.$ [i]2011 Ritsumeikan University entrance exam/Science and Technology[/i]

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

Evaluate \[\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx\]

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 $\int_{-1}^1 f(x)^2dx\leq \frac 23.$

Answer the following questions: (1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$. Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$. (2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$

Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.

Find $\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.$

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 $x \in (0,1)$

Evaluate $ \int_0^{\pi} \sin (\pi \cos x)\ dx.$

For real numbers $ a,\ b$ such that $ |a|\neq |b|$, let $ I_n \equal{} \int \frac {1}{(a \plus{} b\cos \theta)^n}\ (n\geq 2)$. Prove that : $ \boxed{\boxed{I_n \equal{} \frac {a}{a^2 \minus{} b^2}\cdot \frac {2n \minus{} 3}{n \minus{} 1}I_{n \minus{} 1} \minus{} \frac {1}{a^2 \minus{} b^2}\cdot\frac {n \minus{} 2}{n \minus{} 1}I_{n \minus{} 2} \minus{} \frac {b}{a^2 \minus{} b^2}\cdot\frac {1}{n \minus{} 1}\cdot \frac {\sin \theta}{(a \plus{} b\cos \theta)^{n \minus{} 1}}}}$

Consider a sequence $1^{0.01},\ 2^{0.02},\ 2^{0.02},\ 3^{0.03},\ 3^{0.03},\ 3^{0.03},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ 4^{0.04},\ \cdots$. (1) Find the 36th term. (2) Find $\int x^2\ln x\ dx$. (3) Let $A$ be the product of from the first term to the 36th term. How many digits does $A$ have integer part? If necessary, you may use the fact $2.0<\ln 8<2.1,\ 2.1<\ln 9<2.2,\ 2.30<\ln 10<2.31$. [i]2010 National Defense Medical College Entrance Exam, Problem 4[/i]

Evaluate \[\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx\]

Evaluate $\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx$. created by kunny

Consider a right-angled triangle with $AB=1,\ AC=\sqrt{3},\ \angle{BAC}=\frac{\pi}{2}.$ Let $P_1,\ P_2,\ \cdots\cdots,\ P_{n-1}\ (n\geq 2)$ be the points which are closest from $A$, in this order and obtained by dividing $n$ equally parts of the line segment $AB$. Denote by $A=P_0,\ B=P_n$, answer the questions as below. (1) Find the inradius of $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. (2) Denote by $S_n$ the total sum of the area of the incircle for $\triangle{P_kCP_{k+1}}\ (0\leq k\leq n-1)$. Let $I_n=\frac{1}{n}\sum_{k=0}^{n-1} \frac{1}{3+\left(\frac{k}{n}\right)^2}$, show that $nS_n\leq \frac {3\pi}4I_n$, then find the limit $\lim_{n\to\infty} I_n$. (3) Find the limit $\lim_{n\to\infty} nS_n$.

The parametric equations of a curve are given by $x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)$. (1) Find the maximum and minimum values of $x$. (2) Find the volume of the solid enclosed by the figure of revolution about the $x$-axis.

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 $\lim_{n\to\infty} S_n.$

Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\ dt,\ g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\ dt.$ For a real number $a$, find $\sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}.$

Evaluate $\int_0^1 \frac{\ln (x+2)}{x+1}dx.$