$\int_0^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx,$ with $x=\frac{\pi}{2}-t$
How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?
bilinear form - proof I have to prove that the mapping $f(x,y)={\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{j=1}^{n}}x_{i}y_{j}{f}(e_{i},e_{j})$  is a bilinear form, that is, inter alia, the condition: $f(x+y,z)=f(x,z)+f(y,z)$  I have: $f(x+y,z)=f\left({\displaystyle \sum_{i=1}^{n}}x_{i}e_{i}+{\displaystyle \sum_{k=1}^{n}}y_{k}e_{k},{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=f\left({\displaystyle \sum_{i=1}^{n}}{\displaystyle \sum_{k=1}^{n}}\left(x_{i}e_{i}+y_{k}e_{k}\right),{\displaystyle \sum_{j=1}^{n}}z_{j}e_{j}\right)=\ldots$  and what's next?  Help! Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset $K\subseteq \Omega$. Then by Cauchy Theorem and Morera Theorem, $f$ is holomorphic. Let $f_n$ and $f'$ be the derivatives of $f_n$ and $f$ respectively. Prove that $f_n'\to f'$ uniformly on any compact subset $K\subseteq \Omega$.   How to prove? raw data stationary but still can see trend and seaonality is stl Feed lists answered questions as "unanswered" What's the function like sum() but for multiplication? product()? Python's  function returns the sum of numbers in an iterable.  sum([3,4,5]) == 3 + 4 + 5 == 12   I'm looking for the function that returns the product instead.  somelib.somefunc([3,4,5]) == 3 * 4 * 5 == 60   I'm pretty sure such a function exists, but I can't find it. I'm considering taking a loan on my 401k in order to pay off and close 2 credit cards.  I have approximately 30 years until I consider retiring.  The amount of the loan will be $1700 which is less then 20% of my current 401k balance.  I will continue to put 9% of my paycheck into a Roth 401k, and my employer matches 9%.  The loan will be paid back at 3.25% (to myself) and does have $183 in fees.  There are 2 cards I will pay off and close if I follow through with this plan.  One card is 17.9% interest with roughly a $650 balance.  The other is 23% (ish) with roughly a $1,025 balance.  Both of these cards do have an annual fee associated with them, I used them to rebuild my credit and have a newer, lower interest, no fee card that I will keep active for emergencies.  Would it be advantageous for me to pay off these two credit cards and cancel them with a 401k loan? Is a NVIDIA GeForce with Optimus Technology supported by Ubuntu? I am considering buying a Dell XPS 15 laptop. The laptop has a Nvidia 420m card which should work with Linux but I keep reading that the Nvidia Optimus technology isn't supported on Linux. I am not really interested in switching from Nvidia to Intel to save power but need to know that the Nvidia card will in fact work if I installed Ubuntu.  If anyone has experience using a Nvidia card with Optimus technology or even better the exact laptop in question (Dell XPS 15 with 1GB NVIDIA® GeForce®  GT 420M) it would be great.  A major problem holding people back from adopting Linux is this sort of hardware issue. I am a long term Ubuntu user and supporter who can't afford to make a mistake with a purchase like this. I don't want to spend £500+ to find I have no graphics acceleration because Windows 7 is not an option. What is the best way to deal with very complex forms? I'm trying to give the best experience to users of a very huge system (like a CRM). It has a lot of forms and much of these have a lot of fields. My doubt is about if I should use a single column form (which raises too much vertical scrolling), or a two columns form (which needs less vertical scrolling and uses more horizontal white space) and a last one with four columns (which has few scrolling and uses almost whole white space to try to keep most of the fields above the fold).  It's important to say that our clients hate scrolling, and they always ask why we don't use the horizontal white space when we present a one column form).  So I'm little bit confused about it, because a lot of UX studies and good practices recommend to do not use/avoid multicolumn forms but the clients most of times ask multi column forms.  Our clients are very used to desktop programs and not everybody is very familiar with web based systems.  Below follows some possibilities I would like to share to illustrate an example of a complex form in the system with the three possibilities I mentioned:    In the example above I'm using the one column form (most recommended in UX studies), but with a very long vertical scroll.    In the second example I tried to reduce vertical space by using two columns.    This is the client's favorite one, 4 columns using white space and trying to keep everything above the fold.   The lengths of fields are based on the average of characters amount for each field.   I would like to know your opinion about the best approach to this case. I had this question:    All 3 responses were rated at 1 when the bounty expired. Nothing got selected as the "best answer", and I can't even manually select an answer. I am trying to find duplicate accounts in the system and trying to merge them based on our filter criteria using apex. The issue is I am hitting the governor limit of 150 DML statements while doing this.  We will have to put the merge statement inside the loop as we need the parent and child as the parameters in the merge statement. I don't think we can bulkify Merge like other DML statements due to the above reason. I am not sure if there is any better way to do it and avoid hitting governor limits.  Any help in this regard would be greatly appreciated.  Code that I have tried is included below  global class LoyaltyAccountBatch implements Database.batchable&lt;sObject&gt;{        global Database.QueryLocator start(Database.batchableContext info){           String query;         query = ('SELECT  Id FROM  LoginHistory LIMIT 1');          return Database.getQueryLocator(query);    }       global void execute(Database.batchableContext info, List&lt;LoginHistory&gt; scope){             // Get those records based on the IDs         List&lt;Account&gt; templst1 = [SELECT Id,UniqID from account where createddate = LAST_N_DAYS:4 and UniqID &lt;&gt; ''];             system.debug('templist is '+templst1);               Set&lt;String&gt; insertedRecords = new Set&lt;String&gt;() ;         for (Account a : templst1)  {         insertedRecords.add(a.UniqID);     }                 List&lt;Account&gt; templst = [SELECT Id,UniqID from account where NewUniqID IN : insertedRecords                                   order by NewUniqID,UniqID asc];         // Process records          for(integer i=0;i+1&lt;templst.size();i++)          {             if(templst[i].UniqID &lt;  templst[i+1].UniqID)                     {                 system.debug('ids are '+templst[i].UniqID+'--&gt;'+templst[i+1].UniqID);                 String str = '1'+templst[i].UniqID.substring(1,templst[i].UniqID.length());                 String str1 = '1'+templst[i+1].UniqID.substring(1,templst[i+1].UniqID.length());                 system.debug('str is'+str);                 system.debug('str1 is'+str1);                 if(str.equals(str1))                 {                     Merge templst[i+1] templst[i];                     system.debug('value of i is'+i);                 }              }         }        }          global void finish(Database.batchableContext info){         }   } Last seen not updating when using Android app This isn't directly related to the app, but probably more on the APIs it's using. The last seen on the profile page doesn't seem to reflect that I'm actively using the Android app, or taking actions. I was able to vote on items and the last seen property on the profile page said I was last seen 50 minutes ago, when I was using a computer.  I'm assuming it won't count as a visit for the Enthusiast and Fantic badges either. I don't want someone angrily posting on Meta swearing they used the app for a day, yet they lost the consecutive days and has to prove it by showing they voted for something (we're probably far from this situation since it's alpha...) Keyboard shortcut/command for opening a new nautilus window and raise it What is the shortcut for opening a new window (opening folders). for that I have to go to folder options and right click and then need to click on open new window.  or I have to make focus on the folder icon or open a folder then press  Ctrl+N  for opening a new window is there any direct command spcific to open a new window in ubuntu. Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is... Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n\mathbb Z$. Prove that either every element of $H$ is even or exactly half of its elements are even. (Gallian, Contemporary Abstract Algebra, Exercise 24, Chapter 3.) This is a question about American English usage of the word "already". As a UK resident I don't completely understand when I hear Americans give commands like "Stop it already!" In the UK the word already is not normally used in the imperative mood and the sentence I've just quoted would leave an English person thinking "If you're saying I've already stopped it why are you asking me to stop it again?" Install software from packages, or build it from source? Does the artificer's Enhanced Weapon stack on top of hexblade's Improved Pact Blade? I play a hexblade warlock with an Improved Pact Blade longsword. I like the idea of the eldritch tinkerer, who builds his patron's gifts into his gadgets, which fits in a warlock/artificer multiclass. The artificer's Enhanced Weapon infusion would be very nice with the warlock if it stacks with the Improved Pact Blade bonuses, but I can't tell if it does from the class descriptions. Do these features stack, or do I need to rethink this build? How to fix the weird surface in fluid object Do commuting matrices share the same eigenvectors? In one of my exams I'm asked to prove the following      Suppose $A,B\in \mathbb R^{n\times n}$, and $AB=BA$, then $A,B$ share the same eigenvectors.    My attempt is let $\xi$ be an eigenvector corresponding to $\lambda$ of $A$, then $A\xi=\lambda\xi$, then I want to show $\xi$ is also some eigenvector of $B$ but I get stuck. Checking if a key exists in a JavaScript object? Why would Louise telling the Chinese commander about his wife's last words change his mind? In the future in Arrival, the commander says that he changed his mind about going to war with the aliens because Louise called his private number and told him his wife's dying words. My question is what did Louise say to him (like in English) and why did it change his mind?