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

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

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 \[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]

Given a line segment $PQ$ with endpoints on the parabola $y=x^2$ such that the area bounded by $PQ$ and the parabola always equal to $\frac 43.$ Find the equation of the locus of the midpoint $M$.

Answer the following questions. (1) $ 0 < x\leq 2\pi$, prove that $ |\sin x| < x$. (2) Let $ f_1(x) \equal{} \sin x\ , a$ be the constant such that $ 0 < a\leq 2\pi$. Define $ f_{n \plus{} 1}(x) \equal{} \frac {1}{2a}\int_{x \minus{} a}^{x \plus{} a} f_n(t)\ dt\ (n \equal{} 1,\ 2,\ 3,\ \cdots)$. Find $ f_2(x)$. (3) Find $ f_n(x)$ for all $ n$. (4) For a given $ x$, find $ \sum_{n \equal{} 1}^{\infty} f_n(x)$.

A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

Let $ a$ be a positive real number. Find $ \lim_{n\to\infty} \frac{(n\plus{}1)^a\plus{}(n\plus{}2)^a\plus{}\cdots \plus{}(n\plus{}n)^a}{1^{a}\plus{}2^{a}\plus{}\cdots \plus{}n^{a}}$

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

Evaluate $ \int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx$.

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.

Evaluate \[\int_0^1 e^{x+e^x+e^{e^x}+e^{e^{e^x}}}dx\]

Evaluate ${ \int_0^{\frac{\pi}{8}}\frac{\cos x}{\cos (x-\frac{\pi}{8}})}\ dx$.

In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$. [asy] Label f; xaxis(0,60,blue); yaxis(0,60,blue); real f(real x) { return (x^2)/60; } draw(graph(f,0,53),red); label("$y=x^2$",(30,15),E); real f(real x) { return (x^2)/25; } draw(graph(f,0,38),red); label("$y=2x^2$",(37,37^2/25),E); real f(real x) { return (x^2)/10; } draw(graph(f,0,25),red); label("$y=f(x)$",(24,576/10),W); label("$O(0,0)$",(0,0),S); dot((20,400/25)); dot((20,400/60)); label("$P$",(20,400/25),E); label("$B$",(20,400/60),SE); dot(((4000/25)^(0.5),400/25)); label("$A$",((4000/25)^(0.5),400/25),W); draw((20,400/25)..((4000/25)^(0.5),400/25)); draw((20,400/25)..(20,400/60)); [/asy]