I'm stumped here.The interval of integration would make me think to look for symmetry, but it's clearly not even or odd. I asked wolfram alpha to get any ideas, and it gave me a complex number as the value of the integral, which makes no sense. The graph seems to enclose a finite area in the interval, making me think it has a real finite answer. How do I solve this?
As the title states, I'm having quite a lot of trouble with the integral:  $$\int_{-\ln 2}^{\ln 2}  \frac{(\sin x+x)^{1/3}}{e^x}dx$$  The problem is that my standard (admittedly sparse) repertoire of tricks seems to have no effect at all. I can't think of any clever substitution that would work, and it looks like a nasty integral all-round.   I was hoping to apply some symmetry arguments (especially considering the limits of integration), but the exponential function is neither even nor odd, so I don't see how this would work.   I'm afraid I have absolutely no idea how to approach this.
This problem is from a practice exam I was working on.     What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ?   Thoughts. If I find a ring that is easier to handle then this then I can go from there. So I think this is isomorphic to $\mathbb{Z}_2[x]/(x^2-3,x+2)$. And then I am stuck.  Am I correct so far? What should I be looking to do from here?  Thanks for your time and your answers. My online friend is asking for money in order to visit my home country. Is this a legit request or a scam? On Travel.SE we often see questions about a person (usually a love interest) asking for money to pay for airfare, visas, permits and other expenses. For example        What general advice can be given in such situations to the person asking the question? Is it all a scam or can there be legitimate situations in which an online friend could be asking for money? I want to play a game in an older update, but can't see a way to choose a specific version. Is this possible? Use of comma separating two adjectives If I had the phrase       two blond haired, blue eyed people    would the comma be out of place?    should I hyphenate "blond haired" and "blue eyed"? What are the best practices for battery longevity? How do you maximize the cycle count of your battery?  There seems to be a debate between:   Leave plugged in 24/7 but do a full cycle once a month Charge to ~100%, drain to ~10%, repeat.   NOTE: Taking your   Which method is better and why? How much of a difference does it really make?  Here are some of the better sources I've found about battery behavior so far. Even still I feel they don't conclusively answer the question above. See my elaborations below:           Elaborating on Option 1  Now IF Apple engineers were smart and optimized for AC draw when plugged in to not use up your charge cycles, then it seem logical to assume that you are using effectively none of your 1000 charge cycles while plugged in. If that's the case, it would seem that the LiPo drop from holding at 100% would be vastly outweighed by the fact you're not using any charge cycles.  However, this is a purely speculative assumption and I could find no evidence to either confirm or deny.  If this is NOT true, and the AC adapter only goes to charging the battery, then there would be no difference. Your battery would drain to 99%, then back up to 100%, then back down to 99%, etc. Those micro-charge cycles would add up at the same rate as 100% -> 0% -> 100%, and you would get no gain. In this case the negative effect of holding LiPo batteries at 100% would outweigh everything else.  Elaborating on Option 2  There are lots of good reasons why Option 2 is the way it is:   Because of known LiPo chemistry issues, holding a 100% charge for a long time causes the battery to degrade Apple specifically recommends not to leave it plugged in all the time  Draining a battery all the way to zero all the time is bad for the cell (Which is why ~10% is used) Heat is a killer, and when plugged in AND charging, you get extra heat that causes damage.   There are a couple reasons why I'm challenging Option 2:   It seems silly that the most intuitive use case for thousands of people that use their macs as primary computers is wrong. If the hypothesis in option 1 is true, you're just burning unnecessarily through your finite charge cycles. It's a pain to have to remember to keep plugging in and plugging out and be worried about whether or not your device is plugged in. Why is a ConcurrentModificationException thrown and how to debug it I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) What are suitable TV sizes and viewing distances for couch surfing on a 1080p LCD HDTV? It is becoming more common for people to explicitly state the pronouns to be used in addressing themselves: he/him, she/her, or they/them. For example, a name tag for a conference might read:  Cory Lopez Some Company She/Her   Usage seems to have settled on the singular they and them as gender neutral pronouns. Alternatives that were once widespread in niche communities, such as sie/hir, seem to have died out.   Whether or not one agrees with this language  shift, the pairing between subject and object case is fixed.  It's difficult to imagine someone who uses "he" as his subject case using anything but "him" as object case, and it's even harder to imagine that such a person would succeed in getting people to use a non-obvious collection of personal pronouns. (And you may see what I did -- flat out assumed that "he" and "his" automatically go together.)  What are the roots of the practice of presenting subject and object case pronouns as opposed to just subject case, subject and possessive, or some other combination?   (Note: I am not interested in a discussion of the gender politics and cultural shifts involved, just the mechanics of how the current representation was chosen.) Complete view of where the PATH variable is set in bash E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable) The novel centers around a man living a comfortable existence in the pretty distant future.  To support himself, he is occasionally asked to enter a box which will copy his entire body, and send it out faster than the speed of light to some developing world, to work.  His copy on this world wakes up and realizes what a big mistake he's made.  The copy does grueling work, feels lonely, and eventually just dies of radiation sickness.  Back on earth, the man never realizes how bad it is out there, and keeps sending out more copies.  I think the story is probably 80s or earlier - not sure if it's a short story, or the beginning of a novel which I lost &amp; never finished. removeClass and fade animation Let $a, b$ be elements of a group $G$. Suppose that a has order $5$ and that $a^3b = ba^3$. I want to show that that $ab = ba$.  Here is what I think:   We know that we have $a^1, a^2, a^3, a^4, a^5 = 1$. So, $a^4 = a^{-1}$ and $a^3 = a^{-2}$ and $a^{2} = a^{-4}$. Now,   $a^3b = ba^3$ then $a^{-2}b = ba^{-2}$ then $ba^2 = a^2b$ then $ba^{-4} = a^{-4}b$ then $a^{-1}b = ba^{-1}$ then multiply on the left and right by $a$ to get $ba=ab$. Is this correct or is there a simpler approach? how to disable system apps without root? How can a question banned user fix their questions if they're deleted? If a user is question banned  and the , how can the user fix this on their own?  Until recently, advice has been to improve their questions, but since they can't see the questions on their question screen, they can't improve them:     Feature request: Allow a question banned user to see their own deleted questions, or warn them that deleting questions could lead to a question ban. The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ (NBHM_2006_PhD Screening Test_Analysis)     The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the   point on the curve farthest from the line $x = y$   how should i solve this problem We have a SQL Server 2012 instance that contains a database with data for many product lines.   We are looking for a way to take subsets of this data into another database on the same instance.  This doesn't have to be continuous as we are looking for ways to make it easy to export backups of the subsets that we can send out to vendors (we have a company that loads this data into their own sql server to produce a product we sell, naturally its huge and we want to trim it down to just the product lines they work with on a regular enough basis) or use to load SQLite DB's for use with offline applications.  ProductDB - the superset of all product line data.  ProductLine1DB - A subset identical in structure to ProductDB, but with data only relevant to product line 1.  ProductLine2DB - A subset identical in structure to ProductDB, but with data only relevant to product line 2.  This can be done nightly, weekly, as often as we need to but not a continuous replication.  We don't want to add in distributor servers or even extra subscriber servers. We would like the replication to happen on the same server with the subscriber databases being on the same server.  Is this a good candidate for transactional replication?  Why or why not?  We obviously have other options to load this data, but it seems like this might be a good case to use something like transactional replication. I've heard its powerful but prone to issues when not setup optimally.  Note: I am not a DBA, simply a developer, but we are currently looking for a new DBA and some of us are taking over duties like this temporarily. So I have a mac computer but i plugged in a windows keyboard in my mac. I am still trying to figure out how to paste commands into a command block. So how can I paste a command? How do I add newlines between lines printed on the command line? Mail logs are incredibly difficult to read. How could I output a blank line between each line printed on the command line? For example, say I'm grep-ing the log. That way, multiple wrapped lines aren't being confused. Navigation bar overlapped by status bar