Limits of integration become equal upon substitution
$\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$
Not enough reputation - wanting to answer/comment I've seen a few questions I'd like to comment on or give a different answer, but seeing how I don't have 50 reputation (I've been a lurker for years, finally decided to participate) do I really need to ask a few questions before I can comment/answer other people's stuff? Or am I missing something here?  Sorry if this has been answered before, but with a few search tries I couldn't find any similar stuff, even tried google just for "50 reputation stackoverflow" but that didn't give me anything either. Install Java JDK and JRE on Debian 8 I installed Debian 8, I would like to install Java JRE and JDK. I use  and it works.  But, I am afraid because it's a script hosted in a repository. I would like to understand why it doesn't work when I put the extract of  in /usr/java/jre1.8.0_73 directory as . I added the path variable with PATH=/usr/local/jdk1.8.0/bin:$PATH export PATH as explained in  but it doesn't work.  Even if I try to install OpenJDK, the package isn't found. I don't understand why it's so complicated to install Java on Debian; it is very simple on Ubuntu. I would like someone give me step-by-step instructions to install it. Responsive Breakpoints? Twitter Bootstrap uses these break points for responsive design:  Large Display: 1200px Default: 980px Tablet: 768px Mobile: 480px  A client said they are using 1280px for their Large Display breakpoint.  Is there an industry standard for these or are they just made up as we go right now? Do I cast the result of malloc? What if I accidentally run command "chmod -R" on system directories (/, /etc, ...) Why does Jack let Walt go? In Breaking Bad S5 E14, Jack not only lets Walt go, but also leaves a barrel of money to him after killing Hank. I don't understand why he would do this. Jack says that, with 7 barrels of money, he doesn't need to cook meth any more, so there is no reason to keep Walt (the witness of the killing) alive, let alone to give him so much money. What's the rationale behind this? Or it's just mercy or respect? When clicking continue on the first Boot Camp screen,  I am presented with a dialog that displays:     The startup disk does not have enough space to be partitioned.   You must have at least 39 GB of free space available.     This is despite "About this Mac" reporting 271GB of free space:    And "Disk Utility" reporting 268GB of free space:    Searching online I found  which suggested it was the result of Time Machine which i recently enabled and suggests to run sudo tmutil thinlocalsnapshots / 999999999999 which I did, it ran sucessfully, and I restarted, yet to no avail.  What is the solution to this instance of Apple's poor engineering?  Running macOS High Sierra 10.13.4 (17E202) "Its" as a Possessive Pronoun Can a PC with the Alert feat alert his friends? How do I find the right size of filters for a lens? Find weak form of linear transport equation I am stuck on the following problem that says:       a) Find a weak formulation for the partial differential equation $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$    b) Show that $u=f(x-ct)$ is a generalized solution of $${\partial u\over\partial t\ }+ c{\partial u \over \partial x\ }=0$$ for any distribution $f$    My Attempt:   I know that in order to find a weak form of a pde, we need to multiply it by a test function, then integrate it. Also, to find a generalized solution, we need to find a weak solution and just multiply it by the Heaviside function.  Let's take any test function $\phi $, then we have (integrating by parts second part of the integral) $$\int_{\Omega } ( {\partial u\over\partial t\ }+ c{\partial u \over \partial x\ })*\phi(x) dx= \int_{\Omega } {\partial u\over \partial t}\phi(x) dx - c\int_{\Omega } u(x,t)\phi'(x)dx$$ where $\phi $ vanishes at boundaries. So, is it the final form or can we proceed further? And how am I supposed to find a generalized solution?  Can someone help me out? Thanks in advance for your time. Geometric meaning of the determinant of a matrix What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.  Thank you very much. Sorry if this is very basic but here's a question.   Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$.      Why do we have   $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|,  $$   where the left-hand side can be thought of as the $2$-norm $\|\mathbf{v}\|_2$ on $L^2(k^n)$?   $\mathbf{General \; case}$: If this is true for $p=2$-norm, I am guessing that this is true for all $p\geq 1$:  $$ \left( \sum_{i=1}^n |v_i|^p\right)^{1/p}\leq \sum_{i=1}^n |v_i|.  $$     In fact, does this inequality hold when $p$ is a rational number? Move incoming Outlook emails to folder containing previous conversation I file email conversations in various folders. But every new email in that conversation goes to inbox, so I have to manually move it to the folder that contains the rest of the conversation. Is there a way for Outlook to do this automatically with all new emails?  What I want: Outlook will do the following automatically as long as it's running:   For all unread emails in inbox... ...if email is not the first in the conversation, search for folders where that conversation is stored. If found, move the new unread email to the first* folder it finds (except for Inbox, Trash, or Sent). Optionally: If there is more than one folder containing messages in the same conversation, then do nothing.   **If there is more than one folder containing messages in the same conversation, then do nothing*. (This extra rule is not necessary if it complicates the solution too much)  Thanks! When should end punctuation go inside quotes? I have been/am being taught that end punctuation should always go inside quotes. For example, you are supposed to write:     Marvin thought it was "awful."   The problem is I do not see how does this make sense. Intuitively, I always wrote:     Marvin thought it was "awful".   as that makes more logical sense — you want a quote to be an exact replication of what somebody else said, so why should you add punctuation inside?  I always thought it made more sense to not touch the quote and add anything after or before if it must be added.  So, why should I put end punctuation inside quotes? How to show $\lim_{n \to \infty} a_n = \frac{ [x] + [2x] + [3x] + \dotsb + [nx] }{n^2} = x/2$? Using pgfplots, why do I get "undefined control sequence" when trying to use a foreach variable as a parameter to addplot? Since I have multiple plots with error bars, I would like to shift the error bars slightly to make the overall plot more readable. However, when I try to do this using the variable of a \foreach as the amount to shift, I get an "undefined control sequence" error.  \documentclass{article}  \usepackage{tikz} \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture} \begin{axis} \foreach \x/\y in {a/-1cm, b/0cm, c/1cm} {     \addplot+ [         %every error bar/.append style={xshift=\y},         %every node/.style={xshift=\y},         error bars/y dir=both,         error bars/y explicit,     ] coordinates {         (0, 0) +- (0, 1)         (1, 0) +- (0, 1)         (2, 0) +- (0, 1)     }; } \end{axis} \end{tikzpicture}  \end{document}   (The document does not compile when I uncomment either one of the two comments.)  What's wrong here?  EDIT: Added another variable to the \foreach. In my actual use-case it would contain the filename to plot data from or the label to use for the curve. I accidentally deleted a file from my laptop. I'm using Fedora. Is it possible to recover the file? How secure is Activation Lock on iOS 7 (and later)? Smallest Width for Galaxy S and Galaxy S2