Evaluate $\int x^3 \sqrt{4-x^2} dx$  Is my solution correct? In the book, some other (final)solution is given.
$$\int x^3\sqrt{4-x^2}dx$$ I tried changing the expression like this: $$\int x^3\sqrt{2^2-x^2}dx=\int x^3\sqrt{(2-x)(2+x)}dx$$ And that's where I got stuck. I tried integration by parts but it doesn't simplify the expression.
How do I perform a biographical interview? I have a couple of elderly relatives who have not kept a journal and refuse to write down their "life story." I really would like to know their stories and who they were and who they are. They have their favorite stories that they rehash every time I see them, but I'd like to get more. I have no experience with "interviewing" and am looking for information on how to get their stories. Any websites, books, and or tips\techniques would be greatly appreciated. Number of equivalence relations on a set If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it? How do I make animations a steady speed? When I make an animation, it always starts out slow, then speeds up, then slows down at the end.  I cannot figure out how to fix this. Why is ++i considered an l-value, but i++ is not? Why is ++i is l-value and i++ not? How to apply for a UK standard visitor visa as a family? I am an Indian citizen residing in the United States. My wife and I are planning to travel to the UK for a week. We plan to apply for the UK standard visitor visa from the US. I was filling out the application and I have a few questions.   Travel History - The application asks for travel history to the United States, Canada, Australia and New Zealand. Considering that we're resident aliens in the United States, should we consider all the times we've traveled back to the US from India? Travel History - Should we also list out the times we've traveled to India from the US? Cost of the trip - Hypothetically if we plan to spend a total of ~4,000 USD on this trip, should I split up the cost and write down 2,000 USD on my application and 2,000 USD on my wife's application? Income - From what I've read on the forum, my wife must say that "someone else" i.e. me - is paying for her costs and mention my GWF number in the remarks on her application. Would that be sufficient considering that I'll be sending all income documents with my application? Savings - We have joint savings accounts. Should I mention the same amount on both our applications? Posting the documents - Is the up-to-date advice to send all documents in one packet? Or would sending them in separate packets at the same time be better? My thought was to annotate my wife's packet with my GWF number if we were to send them separately. Bank statements - We get e-statements. Would printing those out be enough? How to diff a commit with its parent? Aside from writing an alias or script, is there a shorter command for getting the diff for a particular commit?  git diff 15dc8^..15dc8   If you only give the single commit id git diff 15dc8, it diffs that commit against HEAD. Relationship between Partial Harmonic Sum and Logarithm. $e^{i\theta}=\cos\theta + i\sin \theta$  $e^{i\sin^{-1}x}=\cos(\sin^{-1}x)+i\sin(\sin^{-1}x)$  $i\sin^{-1}x=\ln|\sqrt{1-x^2} + ix|$  $\sin^{-1}x=-i\ln|\sqrt{1-x^2} + ix|$  Now from here I'm kind of lost, since it seems like this should be the definition, but when I look it up, the definition of inverse hyperbolic sine is:  $\sinh^{-1}x=\ln(\sqrt{1+x^2} + x)$  So although they're very similar, I guess I just don't know how to handle the logarithm and anything to the ith power or drop off the absolute value. If anonymity is so important, why is it not universal? A frequent argument brought up in support of voting anonymity is that it can lead to potential grudges. Certainly the review system is of the same nature.  I cannot wrap my mind around the fact that voting is anonymous and reviewing is not. Either both should be public--and this would be fine since public reviewing has been successful--or both should be anonymous--and I assume this would be the desired option since this community holds anonymity so dear.  One could say that voting rates the posts, the actual meat of the site, while reviewing is based on clarifying the purpose of the site and cleaning it up.   answer makes the point that the community can trust that votes are a good measure of quality for the very reason that they do not know who cast them; the mentality that a high-ranking member may have supported something could blur the line between "science" and "politics/religion."  Reviewing, then, can be seen as a judgment of specific members rather than quality of edit/questions, which seems to fall precisely under the "politics/religion" umbrella--the review system is much more vulnerable to grudges due to this.  So let's discuss the differences and the implications of these disjointed anonymity policies. double movement in Scrabble Can you resolve a dispute between myself and my girlfriend? She recently won a game with what I think is a double movement (although there's no mention of this in the rule book).  She put one two letters down to make one word, which also scored linked to another letter. But she then added another letter immediately to the left of the finished word, which then extended the second word. I just don't think you can make a second movement like this in scrabble!    My sister placed down W, D and E, the ones that I circled in red. Is she allowed to do that? I told her it was impossible is it?  Can you resolve this for me please? How do I prevent "Read more: www.site.com" from being injected with non-Tynt JavaScript into copy-paste actions? There are many sites that insert something like "Read more: www.site.com" into your clipboard when you copy text in your browser.  I find this abominable.  A google search reveals   to stop this if the website happens to be using the service  (by just blocking the Tynt domain), but many websites use home-grown JavaScript.  Is there a general way to block this behavior besides just turning off JavaScript?   I use Chrome but I am interested in all solutions.  EDIT: To be clear, I want to be able to keep the rest of the JavaScript functionality for these websites, since many will break without it.  EDIT #2: Here are two example websites which continue to harass me despite my use of a Tynt blocker:  |   Here are two   and two   which explain ways to implement this shady practice manually without Tynt.  Here is a  describing how hard this turns out to be to block reliably.  Here is the  I could find (March 2013) which offers suggestions on how to block this in Chrome using AdBlock, but it didn't work for me. I used the prcomp() function to perform a PCA (principal component analysis) in R. However, there's a bug in that function such that the na.action parameter does not work. ; two users there offered two different ways of dealing with NA values. However, the problem with both solutions is that when there is an NA value, that row is dropped and not considered in the PCA analysis. My real data set is a matrix of 100 x 100 and I do not want to lose a whole row just because it contains a single NA value.  The following example shows that the prcomp() function does not return any principal components for row 5 as it contains a NA value.  d       &lt;- data.frame(V1 = sample(1:100, 10), V2 = sample(1:100, 10),                        V3 = sample(1:100, 10)) result  &lt;- prcomp(d, center = TRUE, scale = TRUE, na.action = na.omit) result$x                                # $ d$V1[5] &lt;- NA                           # $ result  &lt;- prcomp(~V1+V2, data=d, center = TRUE, scale = TRUE, na.action = na.omit) result$x   I was wondering if I can set the NA values to a specific numerical value when center and scale are set to TRUE so that the prcomp() function works and does not remove rows containing NA's, but also does not influence the outcome of the PCA analysis.  I thought about replacing NA values with the median value across a single column, or with a value very close to 0. However, I am not sure how that influences the PCA analysis.  Can anybody think of a good way of solving that problem? Restate theorem in separate file I state a theorem in the main body of a document and prove it in the appendix. I use thmtools's restatable environment to restate the theorem there. However, I now need to have my appendix in a separate pdf file (hence a separate tex document). I'm using xr and xcite to get the cross-references and bibliographical references to play nicely.  But the commands created by restatable are not carried. Is there any way to to the restating of the theorem in the appendix in a separate document?  Here's a MWE:  main document: main.tex  \documentclass{article} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  \begin{restatable}{theorem}{thmcmd}\label{thmlabel} Here is the statement of the theorem. \end{restatable}  The above is Theorem~\ref{thmlabel}.  In this way I can restate it: \thmcmd*  \end{document}   appendix: appendix.tex  \documentclass{article} \usepackage{xr} \externaldocument{main} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  I can refer to Theorem~\ref{thmlabel}.  But I cannot restate it like this: % \thmcmd*  \end{document} I read a book that answers one of my questions, and I would like to quote it as an answer. Is that illegal?    Thanks. When can we interchange “who/which" and "that”? So far I thought the words who and that are usually interchangeable:   There is the man who saw me.  There is the man that saw me.    Both sentences seem fine, who can be replaced with that. But in the following examples:   This is the man to whom I spoke.  This is the man to that I spoke.    The #2 sounds odd, which is why I don’t think it might be correct. The same observation for the words which and that, while   This is the ball which we used today.  This is the ball that we used today.    Seem to be in order and which can be replaced with that. But in the following examples the #2 sounds odd again.   This is the ball with which I scored.  This is the ball with that I scored.    I wonder if there’s any rule regarding the interchangeability of the above mentioned words. Is the "omniscient-omnipotent-omnipresent" definition of God consistent? is commonly defined as an  (infinite knowledge),  (unlimited power),  (present everywhere) entity.  Is there any logical  in this definition?  I have seen several paradoxes like below     Does God know what he's going to do   tomorrow? If so, could he do something   else?" If God knows what will happen,   and does something else, he's not   omniscient. If he knows and can't   change it, he's not omnipotent.      "Can 'an omnipotent being' create a   stone so heavy that it cannot lift   it?"   Do they mean that the defintion of God ( as commonly held) is logically inconsistent? How to pad multiple strings when writing to console so that part is aligned left and part is aligned right? Looking for succinct way of describing the act of solidifying an idea into reality Looking for a succinct way of describing the act of solidifying an idea into reality. "Building [idea] out" seems too generic but I'm trying to avoid a long winded, step by step explanation of the process.  Any ideas would be greatly appreciated. Thanks! AngularJS leaves comments in HTML: is it possible to remove them? I read a novel when I was 12 or 13 (late 80's, early 90's) about two timelines combining on Earth.  It was a full length novel for adults that I swiped from my mom (she doesn't remember it at all.)  I've been trying to figure out what the title is for some time but my google-fu is unable to find the right search terms.    The book was published in the 80's or perhaps early 90's.  The story: It was set on Earth and featured two timelines running concurrently (in the 80's?)  Timeline 1:  The nuclear missile crisis had gone badly and a low level nuclear war had more or less destroyed the US, but small towns still survived but are slowly fading away.  Much of the story was about the scarcity of goods, and the protagonist in the main thread of the book started the story by finding a 6 pack of Coke buried in a ruined house.  Throughout the story people were finding things that had grown scarce (deer, bullets, human beings.)  One strange scene I recall involved mutant/radioactive sweet potatoes that had driven a man insane.  Eventually people in this timeline began having dreams of their lives in Timeline 2.  Some of the people were not having dreams, and it was eventually discovered that they had died in Timeline 2.    Timeline 2:  There is panic in a world recognizable as our 1980's as nuclear war draws near.  Since more and stronger missiles have been developed, the end of humanity is at hand.  Things start disappearing from that world as the war begins.  Timeline 2 was mostly seen as vignettes of panicked people running around.  Combination of Timelines: In a weird, incongruous, almost surely poorly written scene, Gaia (yes, Earth itself) is seen to be combining a dying timeline with a doomed timeline in order to save humanity.  At the end of the novel, kids from Timeline 2 are transported into Timeline 1 in order to repopulate.  The new kids drink all the Coke, in case you were wondering...  Any help finding the title would be welcome.  I recall enjoying the book a lot when I was a kid, but I suspect my mom traded it in at our local used book store because at some point it just disappeared.