I am having trouble in evaluating $\int_{0}^\infty \left(\frac{\log(x^2+1)}{x^2+1}\right)dx$. I proceeded by taking  $\int_{C}\left(\frac{\log(z+i)}{z^2+1}\right)dz$. I don't know how to draw contour here and my attempt lies in that. I could show that $\lim_{R\to\infty}\int_{C_R}\left(\frac{\log(z+i)}{z^2+1}\right)dz=0$, but I couldn't show that $\lim_{r\to 0}\int_{\tau_R}\left(\frac{\log(z+i)}{z^2+1}\right)dz=0$.Please help me through this problem. I'll appreciate any help towards this. PS: $C=[R,r]\cup \{-\tau_{r}\} \cup [r,R] \cup C_{R}$, [considering counterclockwise sense] the branch cut that I used was $[-\pi/2,3\pi/2]$
How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex plane, excluding the negative real axis, but had trouble calculating the residue at $z=i$ (perhaps there is a way of doing this that I don't know of). After that didn't work, I tried a rectangular box integral from $\epsilon$ to $R$, from $R$ to $R+i/2$, from $R+i/2$ to $-S+i/2$, from $-S+i/2$ to $-S+i\epsilon$, from $-S+i\epsilon$ to $-\epsilon+i\epsilon$ and finally a semicircle around the origin, radius $\epsilon$.   Any help would be appreciated.
Variance of a bounded random variable Suppose that a random variable has a lower and an upper bound [0,1]. How to compute the variance of such a variable? What do the X and Y axis stand for in the Fourier transform domain? I have a document divided into \part \chapter \section and \subsection. I have 3 parts and in each part I reset all the counters.  I want that, when I do a crossref to something that is in the 1st part, chapter 3 and section 5, that LaTeX prints something like I-3.5, but  I don't want that in the tableofcontents part that LaTeX prints the part number. For example, what I want is:     Part I      chapter 1 section 1.1 section 1.2      chapter 2   AND NOT     chapter I-1 section I-1.1 section I-1.2      chapter I-2   The command that I've tried   \renewcommand{\thechapter}{\thepart-\arabic{chapter}} \renewcommand{\thesection}{\thechapter.\arabic{section}}   does exactly what I've said  Can anyone help me? Calling the base constructor in C# I have this annoying problem in Gmail. Spammers are by-passing the Gmail spam filters and are so clever that I am not able to create a manual filter of my own to block them. They are sending me junk mails from e-mail IDs of the form:  Jewelskart@greatestjewels.in via watchstore.cleanmail.in  Shoppingkart@shoppers.in via watchstore.cleanmail.in   In general,  something@somethingelse.in via watchstore.cleanmail.in   Now, I cant set a filter for each and every mail as they are from different IDs. The common part is the via address which is watchstore.cleanmail.in. But when I try creating a filter with watchstore.cleanmail.in in From field, Gmail doesn't list these emails. In short, the filter is not able to detect via addresses.  Report Spam/Unsubscribe options aren't working.  How can I get rid of these annoying spammers? What methods of ‘clearfix’ can I use? Kali Linux: apt update returns "Hash Sum mismatch" error I'm experiencing exactly the same issue as described in this question: . Before you mark this as a duplicate however, I have tried the solutions posted there, as well as on numerous other sites, including:  sudo apt-get clean sudo rm -rf /var/lib/apt/lists/* sudo apt-get update   Editing /etc/apt/sources.list with alternate official mirrors, such as  deb http://mirrors.ocf.berkeley.edu/kali kali-rolling main non-free contrib or deb https://http.kali.org/kali kali-rolling main non-free contrib  Everything worked after I first imported the VM. I ran sudo apt update and it found some ~650 packages to upgrade. I ran sudo apt upgrade and it encountered an error partway through. That error was solved using sudo apt --fix-broken install, but that is when this hash sum error began. Unfortunately due to hours of troubleshooting I no longer have the details of the earlier error, but I believe it was an error extracting a package due to corrupt data.  I've tried multiple official mirrors, but I get the same error. Additionally, when I downloaded the Packages.gz file  on my Windows machine (VM host) and computed the SHA256 hash, I got the exact hash that apt printed as the expected value. This led me to believe that the error was not with the mirror but with my VM.  The next thing I tried was wget https://mirrors.ocf.berkeley.edu/kali/dists/kali-rolling/main/binary-amd64/Packages.gz followed by sha256sum Packages.gz, which provided yet another different hash output.  To be clear, I have seen 3 different hashes for the same file:   The "correct" hash shown by apt as expected, which is the one that windows also produced after downloading the file using a browser The incorrect hash calculated by apt, which led to the error A different hash calculated by sha256sum after downloading the file using wget using the same URL as for the browser download   I should also note that I have only been referencing the SHA256 hash in each step. The other hash functions are also mismatched when I run sudo apt update, but the file size is the same. I had considered that downloads might be failing due to limited disk space (it is a VM after all) but I don't think that's the case.  What am I missing? Give each NodeJs Process a name in Windows TaskList What good are free groups? In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count. Modify all bash commands through a program before executing them I am trying to create a program which require such feature. The flow will be like:   User enter a bash command User hit the enter My script will get command, current directory,.. as variables. Program can optionally modify the command. Modified command will get executed normally.   Is there any way to do this?  Note: I need this for my personal use, I am not going to distribute this program. The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that:     Lemma: If $(D,g)$ is a Euclidean domain such that its invertibles are $\{1,-1\}$, and $x$ is an element of minimal degree among the elements that are not invertible, then $D/(x)$ is isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.   Now the exercise asks:     Prove that $A$ is not a Euclidean Domain.   Everything hints to an argument by contradiction: let $(A, d)$ be a ED and $x$ an element of minimal degree among the non invertibles we'd like to show that $A/(x)$ is not isomorphic to $\mathbb Z/2\mathbb Z$ or $\mathbb Z/3\mathbb Z$.  How do we do that? My problem is that, since I don't know what this degree function looks like, I don't know how to choose this $x$!  I know that the elements of $A/(x)$ are of the form $a+(x)$, with $a$ of degree less than $x$ or zero. By minimality of $x$ this means that $a\in \{0, 1, -1\}$. Now I'm lost: how do we derive a contradiction from this? Number of Pairs of Subsets Find the number of Pairs $(A,B)$ of subsets of $[n]$ such that $A ⊆ B$? I just need clarification in my thought process. My professor's wording at times can through me off. I just want to know if I am on the right track. We will have $3$ regions: 1.Outside $B$ 2.Inside $A$ 3. Inside $B$, Outside $A$ I believe the number of pairs is $3^n$ but I am not sure. Any help would be appreciated. Minecraft directory doesn't have a 'Resource Packs' folder? I go into my .minecraft folder and there's Assets and all the other files and folders, but no resource packs folder?  If I create one manually, and place the textures inside, they don't show up in-game.  I'm on a Mac. Word for a person who continually tries to teach you when it's not necessary I've experienced this way too many times, and the best example I can give is a person who treats you like you have no idea how to do X and who keeps teaching you how to do it. I feel handicapped. Take for example how to play a card game. You know how to play the game because you've played it 5 times. However, there is a person always insists on explaining the game to you - like you are always new to the game - when you don't actually need any help and you just want to think for yourself. Another example situation is doing a repetitive task which you already have experience with, and yet the person continually tries to 'teach' you when you don't need help/already know how to do it. It almost makes you mad because you really don't need help. As an example which recently happened to me:  &quot;Hey, do you remember how to start the wood chipper?&quot; &quot;Yes I do&quot; &quot;Okay, well, let me show you one more time&quot;  Again, is there a term for someone who does this? I may not be coming in tomorrow...    I might not be coming in tomorrow...       When should I use "may" and "might"? Kernel of composition of linear transformations Let $f : U \to V$ and $g : V \to W$ be linear transformations on the vector spaces $U$, $V$, and $W$. Supposedly,  $$ \dim(\ker(g \circ f)) = \dim(\ker(f)) + \dim(\ker(g) \cap \operatorname{im}(f)). $$  How might I go about proving that?    (Attempt:) The $\dim(\ker(g)\cap\operatorname{im}(f))$ term suggests to me that I should define a vector space $V' = \ker(g) + \operatorname{im}(f)$ to invoke the theorem that  $$ \dim(V') = \dim(\ker(g)) + \dim(\operatorname{im}(f)) - \dim(\ker(g) \cap \operatorname{im}(f)), $$  but I don't really see where to go from there. Examples of PCA where PCs with low variance are "useful" Aluminum foil is said to be not absorbing light at all. It reflects light. So, does it mean that a more shiny aluminum foil will reflect more light and thus make the room more cooler as compared to less shiny foil?  If yes, then where does unreflected light go in case of less shiny aluminum foil?  Does less shiny and more shiny actually make a difference? Why?  In order to keep whole room cool will it help if I stick the foil to the outside of wooden door (exposed to the sunlight) too? I dont quite understand how pipes work in bash.  I know that it takes an output from one command as the input in another command.  What an output is I can get because it's what the command prints out to the screen.  But how do I know what input a command will take?  Here is an example I thought would work:  which gem | rm   Unfortunately it didn't.  Which gem prints out /usr/bin/gem so that must be the output right?  I thought that was given to rm so it woud be rm /usr/bin/gem but I was wrong.  So my question is, how do I know what input a command takes? 2016: a year in closing