How can we calculate the Integral $\displaystyle \int \ln\left(1+2a\cos x+a^2\right)dx\;,$ where $a\in \mathbb{R}.$  $\bf{My\; Try::}$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2},$ we get ...  \begin{align}\displaystyle \int \ln(1+2a\cos x+a^2)dx &amp;= \int \ln\left(1+2a\cdot \left(\frac{e^{ix}+e^{-ix}}{2}\right)+a^2\right)dx\\ &amp;= \int \ln(1+a\cdot (e^{ix}+e^{-ix})+a^2)dx\\&amp;=\int \ln \left((1+ae^{ix})\cdot (1+ae^{-ix})\right)dx\\  &amp;= \int \ln(1+ae^{ix})dx+\int \ln(1+ae^{-ix})dx\end{align}  Now How can I solve after that  Help me  Thanks
For $a\ge 0$ let's define $$I(a)=\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right)dx.$$ Find explicit formula for $I(a)$.   My attempt: Let  $$\begin{align*} f_n(x) &amp;= \frac{\ln\left(1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2\right)-\ln\left(1-2a\cos x+a^2\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(\displaystyle\frac{1-2 \left(a+\frac{1}{n}\right)\cos x+\left(a+\frac{1}{n}\right)^2}{1-2a\cos x+a^2}\right)}{\frac{1}{n}}\\ &amp;=\frac{\ln\left(1+\dfrac{1}{n}\left(\displaystyle\frac{2a-2\cos x+\frac{1}{n}}{1-2a\cos x+a^2}\right)\right)}{\frac{1}{n}}. \end{align*}$$ Now it is easy to see that $f_n(x) \to \frac{2a-2\cos x}{1-2a\cos x+a^2}$ as $n \to \infty$. $|f_n(x)|\le \frac{2a+2}{(1-a)^2}$ RHS is integrable so $\lim_{n\to\infty}\int_0^\pi f_n(x)dx = \int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2} dx=I'(a)$. But  $$\int_0^\pi \frac{2a-2\cos x}{1-2a\cos x+a^2}=\int_0^\pi\left(1-\frac{(1-a)^2}{1-2a\cos x+a^2}\right)dx.$$ Consider $$\int_0^\pi\frac{dx}{1-2a\cos x+a^2}=\int_0^\infty\frac{\frac{dy}{1+t^2}}{1-2a\frac{1-t^2}{1+t^2}+a^2}=\int_0^\infty\frac{dt}{1+t^2-2a(1-t^2)+a^2(1+t^2)}=\int_0^\infty\frac{dt}{(1-a)^2+\left((1+a)t\right)^2}\stackrel{(*)}{=}\frac{1}{(1-a)^2}\int_0^\infty\frac{dt}{1+\left(\frac{1+a}{1-a}t\right)^2}=\frac{1}{(1-a)(1+a)}\int_0^\infty\frac{du}{1+u^2}=\frac{1}{(1-a)(1+a)}\frac{\pi}{2}.$$ So $$I'(a)=\frac{\pi}{2}\left(2-\frac{1-a}{1+a}\right)\Rightarrow I(a)=\frac{\pi}{2}\left(3a-2\ln\left(a+1\right)\right).$$  It looks too easy, is there any crucial lack?  $(*)$ — we have to check $a=1$ here by hand and actually consider $[0,1), (1,\infty)$ but result on these two intervals may differ only by constant - it may be important but in my opinion not crucial for this proof.
BASH script to set environment variables not working I've written the following script to set some environment variables when needed.  #!/bin/sh export BASE=/home/develop/trees echo $BASE export PATH=$PATH:$BASE echo $PATH   Below the command and the results I can see on my terminal: the script runs, but the variables are not set at the end.  ~$: ./script.sh /home/develop/trees /bin:......:/home/develop/trees ~$: echo $BASE  ~$:    What's wrong? Thanks in advance. Mirko biblatex: \parencite in note field prints spurious ; with authoryear-icomp style I am following Carroll's GR book. He explain that it is convention to parameterize geodesics of photons by a parameter $\lambda$ such that  $$p^\mu ~=~ \frac{d x^{\mu}}{d \lambda}.\tag{3.62}$$  But this is the definition of 4-velocity for a massive particle in the case of $\lambda=\tau$ equal to proper time. My question then, is $$u^\mu = p^\mu$$ for all massless particles? I ask because it's well-known that proper time $\tau$ freezes for massless particles. Spartan 6 DCM unstable clock output The link here refers to the clocking resources of spartan-6 FPGA. I am using the DCM-CLKGEN primitive described in the link, to generate a 8x clock based on an input clock. It works fine as long as the input clock is stable.  But now i sweep the input clock at every 90 us by ±5%. I see that although the DCM does not lose lock, it takes very long (15 to 20 us) to change the output freq after changing the input frequency. This is for me undesirable. Any suggestions?  This question was posted on the xilinx forum and the suggestion was to reset the DCM everytime it loses lock. But the problem is that the DCM is not losing lock  Also, this was earlier posted on stackoverflow and now has been reposted here. stop Ipod from asking for old Apple ID password Concatenating a string and primary key Id while inserting How to define servlet filter order of execution using annotations in WAR Inequality $\sum\limits_{1\le k\le n}\frac{\sin kx}{k}\ge 0$ (Fejer-Jackson) Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$, $$ \sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0. $$  I have this question a very long time ago from a book or magazine but I cannot solve it by myself and did not know how to solve it until today.   My try: for $n=1, 2, 3$, one can check this by hand. Proof that a sum of Bernoulli rvs has Binomial distribution Let $\{X_n, n \geq 1\}$ be iid Bernoulli random variables with $\Pr(X_1 = 1) = p = 1 - \Pr(X_1 = 0)$ and let $S_n = \sum\limits_{i=1}^n X_i$ be the number of successes in $n$ trials. Show $S_n$ has a binomial distribution by the following methods:  1.) Prove for $n\geq0,$ $1\leq{k}\leq{n+1}$  $$\Pr(S_{n+1} = k) = p\Pr(S_n = k - 1) + q\Pr(S_n = k)\text{ where }q = 1-p.$$  2.) Solve the recursion using generating functions.  For the first method, I think I need to use convolution of $n + 1$ Bernoulli rvs, but I cannot immediately see the progression. For the second method, I don't exactly understand how to proceed either. This is problem 1.2 from Resnick's Adventures in Stochastic Processes. After installing Ubuntu Studio 12.04, I have found that it uses a low latency kernel.   I searched for an explanation and how to change back it to a real-time or generic one. However, it looks like this part of Linux hasn't been covered to explain the details.   Q: Why choose a low latency kernel over a generic or real-time one?  PS: I have already read the answers from  question and  post. Is it possible to add non-number characters on the left of a code block?  Specifically, I want to add ... to correspond with some source code (see image).    I am currently using commands to stop / start the line numbers. Here is the code:  \documentclass[12pt]{article}  \usepackage[a4paper, margin = 20mm, top = 30mm, bottom = 25mm]{geometry} \usepackage{xcolor, soul} \usepackage{listings}  \definecolor{codebackground}{rgb}{0.95, 0.95, 0.92}  \lstset {     frame            = tb, % hrule above and below     keepspaces       = true,     columns          = flexible,     basicstyle       = {\normalsize\ttfamily},     escapeinside     = {(*@}{@*)}, % for escaping     backgroundcolor  = \color{codebackground},     showstringspaces = false,     language         = C,     keywordstyle     = \color{blue},     stringstyle      = \color{red},     commentstyle     = \color{teal},     numbers          = left, % {none, left, right}     firstnumber      = 1,     numberstyle      = \scriptsize\color{black},     numbersep        = 5pt }  \let\origthelstnumber\thelstnumber \makeatletter % for stopping line numbers \newcommand*\stopnumber {     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\relax         \advance\c@lstnumber-\@ne\relax     } } % for starting line numbers \newcommand*\startnumber[1] {     \setcounter{lstnumber}{\numexpr#1-1\relax}     \lst@AddToHook{OnNewLine}     {         \let\thelstnumber\origthelstnumber         \refstepcounter{lstnumber}     } } \makeatother  \begin{document} \begin{lstlisting}[firstnumber = 456] int third_function(const int c) {     /* error checking */     if(c &lt; 0)     {         return;     } (*@ \stopnumber @*)      ...     (*@ \startnumber{512} @*)     /* other stuff */     int d = 4;     char* word = "third function"; (*@ \stopnumber @*)      ...     (*@ \startnumber{678} @*)     return (c * d); } \end{lstlisting} \end{document} Are files created in "/storage/emulated/0" supposed to appear in "/storage/sdcard0"? From what I guess, internal storage is available by every user. If emulated storage is a storage location, made for a specific user, are files created in /storage/emulated/0 supposed to appear in /storage/sdcard0? I assume no, because files that are user created should show up in emulated only, otherwise files created by the user would be available for all other users to see. I came to know about , it's a website that allow search for flights by country. I mean you just enter from like "Germany" and to like "Belgium" and you get best offers from all airports between the two countries. That is great. And I was able to arrange a good-price trip to Brussels.   Now I want to arrange another trip to Turkey, but Whichbudget is not helping much this time. And I can't find any other websites that allow "search by country".   Update: Not duplicate to  as his question is broad to the extent of a state. But my question is broad to the extent of a whole country + less price + "Any Time". If anyone checks , he'll understand what I'm saying. Especially that whichbudget allows "Any Time" option, not just +/-3 days.  Does anyone know more similar websites? Solution of limit $\lim\limits_{x\to 0} \frac {e^{-1/x^2}}{x} $ Small question, I'm trying to solve this limit but I just can't wrap my head around this problem.  $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$  L'Hopital just seems to make it messier.   It's probably pretty simple - I'd like to hear what I'm missing. I want to show that if a real orthogonal matrix $A$ has determinant $-1$ then $\lambda=-1$ must be an eigenvalue of $A$.  I have proven this in a long-winded way and I was wondering if these is a quick way of seeing it. Proof a $(2n-1)$-compact manifold I have no idea how prove that $$\{(z_0,\ldots,z_n)\in\mathbb{C}^{n+1} \quad| \quad z_0^d+z_1^2\ldots+z_n^2=0, \quad |z_0|^2+|z_1|^2\ldots+|z_n|^2=2\}$$ is a $(2n-1)$-compact manifold.  How give the charts. For $n=1$ and $d=2$ not distinguish which 1- manifdold is? Is there a direct download link with a raw data dump of Stack Overflow? (not a torrent) I would like to create a local MySQL copy of Stack Overflow for academic purposes (school project). As far as I know, the database dump of the  database is available only in an XML format that I will have to convert to SQL on my own with some tool.  Most of direct download links I found were broken (404). The only non-torrent source that I have found and that is still alive is  but the latest Stack&nbsp;Overflow dump there is from October 2009, content of which looks like this:    Where could I find a direct download link with the Stack&nbsp;Overflow dump?    Note: I am not able to use torrents due to network policies, yet for the completeness of this question, here's the link that might help someone:    Other related questions / links that haven't helped me:     ~ all links there give 404  ~ from September '09 with another broken link  ~ from about a year ago with a clearbits.net link  ~ also not working anymore Algorithm to return all combinations of k elements from n I want to write a function that takes an array of letters as an argument and a number of those letters to select.   Say you provide an array of 8 letters and want to select 3 letters from that. Then you should get:  8! / ((8 - 3)! * 3!) = 56   Arrays (or words) in return consisting of 3 letters each. If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian? I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample. Issue with regex backreference in Python