How to evaluate $$\displaystyle\int \sin^3(x) \,\operatorname{d}x$$  without using integration by parts?
In evaluating integral $\int\sin^{3}x dx $ I am pretty sure we need to use substitution $e^x=t$, but can't go  next step.
indentation of a whole paragraph Why GCP Kubernetes Node host location for europe-north1 seems to be in US? Archimedes' principle for two liquid layers Problem: I have a cylindrical vessel of height $H$ and radius $R$. There are two liquid layers in the vessel. The first has density $D_1$ and height $h_1$, the second has density $D_2$ and height $h_2$. The second liquid is floating on the first liquid (thus $D_2 &lt; D_1$) and they are both entirely within the vessel ($h_1+h_2 \le H$). Now I have a cube of density $D$ and edge length $n$. The cube fits into the vessel ($n \le \sqrt2R$) but it might not be completely inside it ($n&gt;H$ is possible). I place it into the vessel such that one side is horizontal (the cube is not lopsided). Some liquid might overflow as a result. How do I use Archimedes' principle to calculate the topmost liquid level after placing the cube? I can ignore damping and other physical effects. EDIT: Note that the only values I know are $H, R, D_1, h_1, D_2, h_2, n$ and their abovementioned constraints.  My workings: I tried using the formula directly, where $\text{weight of cube}=\text{weight of displaced liquid}$. I first get $Dn^3=D_2n^2s_2$ where $s_2$ is the height submerged in the second liquid. If $s_2 \le h_2$ then the cube is only submerged in the second liquid. So the answer is $\min(H, h_1+h_2+{n^2s_2\over\pi R^2})$, accounting for overflows. This case is simple. But if $s_2&gt;h_2$ then the cube can be submerged in the first liquid too. I thought of subtracting the weight of the second liquid displaced to obtain the height submerged in first liquid, but realize that I don't know that. The weight displaced depends on the final liquid height, which can be affected by overflows, whether the cube fits totally within the two liquids, etc. In fact, it can be the case that after the liquid level rises, the cube now displaces more of the second liquid, causing it to not be submerged in the first (is that even possible?) It seems very messy and I have no idea how to start. :( Does anyone have a nice solution to this problem? "Show desktop" keyboard shortcut in Mac OS X I am using a MacBook Pro with Mac OS X 10.5. Is there a similar function to "show desktop" (shortcut is WIN+D) on a Mac?   It should hide all applications and show a clean desktop. Redshifting of Light and the expansion of the universe I asked a question almost as soon as I got my account. I got a few decent answers, but none of them helped solve my problem. The question has long since become buried by newer questions, and there's still no accepted answer. Unfortunately, because there are quite a few answers with votes, to a casual observer it probably looks like the question was answered and I just neglected to mark an answer as accepted, and the system does not consider it "unanswered".  Or, I asked a question that was a bit more difficult to answer. After waiting for some time, I received no answers, little or no comments, and only a few views. The question does show up in the "unanswered" list, but no one has answered my question.  In either case, what's the protocol for me to try to get this question answered?  I see several options:   Post the question a second time, possibly linking to the original, to get attention for the original question. This is encouraged on sites like Reddit.  However, it's probably not appropriate here, since we're encouraged to check for duplicates and not submit a question that's already been asked by someone else, so this probably falls under that heading. Wait for some kind of wiki-like talk page to be implemented, then post a plea on the talk page for input and hope that activity on the talk page bumps up the question. Edit the question, but do nothing further on the Stack Exchange site. If I edit my post to say that I haven't accepted any of the current answers but am looking for more, then the question might get answered eventually by people browsing through old questions to find something they can answer. Post the question on other sites, but link to my original question on Stack Exchange. People will either answer here, or they'll answer on the other site and I can post the accepted answer here myself.  This might be considered spamming, so it might be better to do this without the link.      For more information, see the "" page in the . Equivalent Definitions of the Operator Norm How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &amp;= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &amp;=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert\leq 1\}\\ &amp;=\sup\{\lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert = 1 \}\\ &amp;=\sup\left\{ \frac{\lVert Av\rVert}{\lVert v\rVert}\;\colon\; v\in V\text{ with }v\neq 0\right\}. \end{align*}$$ How to run a specific program as root without a password prompt? I need to run something as sudo without a password, so I used visudo and added this to my sudoers file:  MYUSERNAME ALL = NOPASSWD: /path/to/my/program   Then I tried it out:  $ sudo /path/to/my/program [sudo] password for MYUSERNAME:    Why does it ask for a password? How can I run/use commands as root with a non-root user, without asking for a password? Why does my Nikon D3100 give the error "Lens not attached" with a Vivitar 200 telephoto lens? I have a Nikon D3100 camera, and am trying to use an "Automatic Fixed Mount Vivitar 200mm lens" with it. When I attach the lens, I get the message "Lens not attached". and nothing works. HELP!  I can't easily take it to a camera shop, as I live in the Mojave Desert, about 200 miles away from any camera shop. This is a new lens, and a new camera. Should I really have been awarded mortarboard on a Per-site meta? The  caused me to win , yey!  But wait.     [Mortarboard] Hit the daily reputation cap for the first time   Per-site metas don't have any reputation... I am confused :-)  I know why I got the badge. Should I have?  Let me rephrase. Is this behavior desirable? Find the limit: $\lim_\limits{x\to 0}{\frac{\left(1+x\right)^{1/x}-e}{x}}$ Thousands of 404 errors in Google Webmaster Tools Because of a former error in our ASP.Net application, created by my predecessor and undiscovered for a long time, thousands of wrong URLs where created dynamically. The normal user did not notice it, but Google followed these links and crawled itself through these incorrect URLs, creating more and more wrong links.  To make it clearer, consider the url     example.com/folder   should create the link     example.com/folder/subfolder   but was creating     example.com/subfolder   instead. Because of bad url rewriting, this was accepted and by default showed the index page for any unknown url, creating more and more links like this.     example.com/subfolder/subfolder/....   The problem is resolved by now, but now I have thousands of 404 errors listed in the Google Webmaster Tools, which got discovered 1 or 2 years ago, and more keep coming up.  Unfortunately the links do not follow a common pattern that I could deny for crawling in the robots.txt.  Is there anything I can do to stop google from trying out those very old links and remove the already listed 404s from Webmaster Tools? As opposed to the generic polynomial form for utilizing the Eisenstein Criterion ($a_nx^n+a_{n-1}x^{n-1}+\dots+a_0\in\mathbb{Z}[x]$ is irreducible in $\mathbb{Q}$) how do we prove that if $p$ is a prime, $x^{p-1}+x^{p-2}+\dots+x+1$ is irreducible over $\mathbb{Q}$? Can you Dash twice on the same turn? The Dash action gives you extra movement for the current turn (equal to your speed). If I have an effect (like the Expeditious Retreat spell or the monk's Step of the Wind feature) which allows me to dash as a bonus action, can I also dash as my regular action, for triple movement? Story about a future with races How do comments work? Across the Stack Exchange network you may leave comments on a question or answer.  How do comments work? What are comments for, and when shouldn't I comment? Who can post comments? Who can edit comments? How can I format and link in comments? My comment doesn't contain some of the text I typed in it; why? My comment appears to be long enough, but I can't post it because it's too short; why? Who can delete comments? When should comments be deleted? Why can't I comment on specific posts even if I have enough reputation to comment? When can comments be undeleted? What are automatic comments? How can I link to comments? Anything else I should know about comments?  Also see the FAQ articles on  and . Background: I am currently employed by a start-up that, so far, is doing really well. As part of my contract, I have stock options that will come available on a typical vesting schedule. A quarter of my options are set to vest later this year. Also, the company is not public yet, and it may be several years before it is public / I could sell my shares of the company.  My question is, is there any reason I should exercise them as soon as they vest? As far as I know they don't expire as long as I work at my company (if I leave for any reason, then I only have 3 months to exercise).  I don't foresee leaving any-time soon, and if I did, I would have 3 months to make a final decision.  Is there any disadvantage to just letting the option stay an option for a while? How to match all occurrences of a regex how to prove $G_1$ and $G_2$ aren't isomorphic? I have been using the default Ubuntu 11.10 with Unity for some time now and decided to start customizing it.  I have already been able to make some changes with the gnome-tweak-tool. However, there are many themes that I get from gnome-look.org that don't appear in the gnome-tweak-tool even after putting them in the /usr/share/themes folder.  Any idea why this is happening? Thanks.