$$\int_{-2}^{2} \frac{x^2}{(1+5^x)}dx$$ I tried taking the substitution of $$1+5^x = u$$ in hope i will get product of two functions which can be integrated by parts but it turns out to be a very complicated function. Is it possible to integrate it using some property of definite integral?
$$\int^2_{-2}\frac{x^2}{1+5^x}$$  How do I start to integrate this?  I know the basics and tried substituting $5^x$ by $u$ where by changing the base of logarithm I get $\frac{\ln(u)}{\ln 5}=x$, but I got stuck.  Any hints would suffice preferably in the original question and not after my substitution.  (And also using the basic definite integrals property.)  Now I know only basic integration, that is restricted to high school, so would prefer answer in terms of that level.
How can I add a bookmark in my nautilus folder explorer Redirect STDERR and STDOUT to different variables without temporary files func() {     echo 'hello'     echo 'This is an error' &gt;&amp;2 }  a=$(func) b=???   I'd like to redirect the stderr to b variable without creating a temporary file.   echo $b  # output should be: "This is an error"   The solution that works but with a temporary file:  touch temp.txt exec 3&lt; temp.txt a=$(func 2&gt; temp.txt); cat &lt;&amp;3 rm temp.txt   So the question is, how do I redirect the stderr of the  function func to the variable b without the need of a temporary file? Keep processes running after SSH session disconnects I sometimes have long running processes that I want to kick off before going home, so I create a SSH session to the server to start the process, but then I want to close my laptop and go home and later, after dinner, I want to check on the process that I started before leaving work. How can I do that with SSH? My understanding is that if you break your SSH connection you will also break your login session on the server, therefore killing the long running process. Georeferencing vector PDF/SVG/DXF linework using QGIS? I want to  create a small web application to visualize election results on a per poll site basis for my city. The result is going to be very similar to what has been done for Berlin . Hopefully this helps in understanding what I'm going for.  The election raw data on the scale I need is published and I know how to visualize this on the grounds of a shapefile with separate polygons lining out the election districts. My problem is - as you could probably guess - that something like this doesn't already exist.  What actually is published though, is a PDF file lining out the borders of the districts I'm interested in:   I'm not keen on redrawing the linework myself, so I'm looking for a way to make a shapefile out of this. Luckily, the PDf includes the linework as separate vector objects, I extracted these into a SVG file:   I already tried converting the SVG to a DXF and importing it into qgis 2.0.1. This actually works, but (as expected) places the vectors in the atlantic ocean and I can't find a way to edit it and fix this.  What can I do now to georeference these vectors with open source tools and finally produce a shapefile? Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46. Oxford Comma Conventions Why is the last digit of $n^5$ equal to the last digit of $n$? I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas about how I can figure out the proof myself? Thanks. I have successfully transferred "Killing Floor" from my brother's laptop to my own via external hard drive. The problem, now that I have all the files, is that I don't know how to launch the game through Steam. All I did was copy files from his Laptop to the hard drive, then from the hard drive to my laptop, then I put the files in the "Steamapps" folder under "Steam" in my laptops hard drive. So the question is, how do I start up the game through Steam? How do I enter a strong (long) DKIM key into DNS? Let $f$ be a nonconstant element of ring $\mathbb Z[x]$. Prove that $\langle f \rangle$ is not maximal in $\mathbb Z[x]$.   Let us assume $\langle f \rangle$ is maximal. Then $\mathbb Z[x] / \langle f \rangle$ would be a field. Let $a \in \mathbb{Z}$. Then $a + \langle f \rangle$ is a nonzero element of this field, hence a unit. Let $g + \langle f \rangle$ be its inverse. Then $a g - 1 \in \langle f \rangle$, hence $ag(x)-1 = f(x)h(x)$ for some $h \in Z[x]$, hence $ag(0) + f(0)h(0) = 1$, thus $(a,f(0))=1$ for all $a \in \Bbb Z$, contradiction, hence the proof.  Is my argument correct? Is there any other method? Recovering the data from a failed WD My Book NAS drive My Western Digital 500GB NAS drive (which is my main backup device) has recently broken. Windows can no longer use it as a network drive (using Map Network Drive), and the web-based control panel is INCREDIBLY sluggish, taking as much as 5 minutes to load a single page.  However, when it does load, it reports the Drive Status as OK, and has the correct disk usage values, which leads me to think the actual drive is OK, and it's the networking components that have failed.  I have checked the wires and router, and so it their isn't another point in the network causing the problem.  Is there any way to get the data back? It's a backup, so I still have about 80% of the data spread out across 3 computers (the only major loss being my music library, and I have the discs for a fair proportion of that).  However, it'd be much more convenient if I can get the data from the drive itself, as it is organised properly on the drive. Onomatopoeia for disgust I'm trying to crack a cypher that I believe is Vigenère encrypted and I'm currently stuck.  I calculated the key length by finding repeated sequences in the cypher and calculating the the common factors.  Word    distance factors agzd    16       2,4,8,16        zna     92       2,4,23,46,92 zdc     54       2,3,6,9,18,27,54 csio    44       2,4,11,22,44 omy     46       2,23,46 zab     5        5   I assume the key-length is 4 as these are the once that are common factors, but from here on trying to map the correct caesar shiffre for each position I'm stuck.  Can somebody confirm that my key-length calculation has no errors which would explain why I can't solve the cypher.  Here is the cypher:  jD 2GSA 7HNA ZNA w.m. V7KXA 36HBM MKE5 /H SGP9 +R GKKJ 2I A4OTC KDBCUR. 19+R 8Y TKJ JG4 IGO6 6U4T OJ hH/BIK. 196 qFKT+9 BNJK ZD6+Q AGZD. kJ'R EaOP6 4NCR ZK 36 / AGZD6C/HOIE2D. x0ZNAC2S8IO8DI HB xX8D4D 0XK O66M 0Y MN62S 8TZABBD2Za8BI. lBJ HA+DF 6UU/ 2J /FOZDC6S8I OJ kDC8G OO 2BLCYZ O66M 0Y G 925F4 UL J2JHCTGH FHH3K I was reading a review of 5e game play  from our RSS Feed. And in it he makes an observation about a strange rule:     Weird rule thing. You recover half your level in hit dice during a   long rest, which means level 1 characters never recover hit dice.   That didn't make sense to me, and so I double checked the long rest rule and it reads:     At the end of a long rest, a character regains all lost it points.   The  character also regains spent Hit Dice, up to a number of dice   equal to half of the character’s total number of them. For example, if   a  character has eight Hit Dice, he or she can regain four spent Hit   Dice upon finishing a long rest.   And of course the rules state that in general you always round down.  Am I missing something? Is there a good explanation for why at first level you can't recover hitdice? I am trying to make a custom food recipe. I'm using spawn eggs with /replaceitem because spawn eggs are not obtainable in survival. This works fine when the player crafts only one spawn egg, but when I scroll over multiple spawn eggs, I receive only 1 of the custom food. BTW this food uses custommodeldata, so I can't simply add it into the crafting recipe. I cannot seem to find a way around this. If I make the item non-stackable, such as a piece of armor, mass crafting would be impossible. If I make an item.dropped scoreboard, the player could keep the item and use it to spawn mobs (griefing or for own profit.)  My command:    execute as @a[nbt={SelectedItem:{id:"minecraft:villager_spawn_egg"}}] run replaceitem entity @s weapon  minecraft:melon_slice{CustomModelData:10000101,display:{ Name:"{\"text\":\"Blue Melon Slice\",\"italic\":\"false\"}"}}     My problem is that the command does not see item count, as it is often used for weapons and armor only.    One way I thought to fix this issue is by storing the amount of custom food crafted in a scoreboard item and then giving the player that same count of the custom food, but I do not know how to do this; consequently, I don't know if this works.    Is there a way to do this method, or is there another, simpler method possible? Preservation of ideals in localized rings Refer to localization of rings in Lang's Algebra p. 108. In particular, let $A$ be a commutative ring, $S$ a subset of $A$ that is a submonoid of the multiplicative monoid structure of $A$, containing the identity. Then  $S^{-1}A$ is the set of elements $\frac{a}{s}$ with $a \in A$ and $s \in S$.  We have $\frac{a_1}{s_1}=\frac{a_2}{s_2}$ if there exists $\bar{s} \in S$ such that $\bar{s}(s_2a_1 - s_1 a_2)=0$. Now consider the injection of $A$ into $S^{-1}A$ given by $x \mapsto \frac{x}{1}$. This is a ring homomorphism. Let $J$ be an ideal of $A$. Ring homomorphisms preserve ideals and so the injection of $J$ into $S^{-1}A$, call it $J^{*}$, will be an ideal. My problem is that i don't see why for any $\frac{a}{s} \in S^{-1}A$ and any $\frac{j}{1} \in J^{*}$ we will have that $\frac{aj}{s}$ will be in $J^{*}$, i.e. why there must exist an element $\kappa \in J$ such that $\frac{aj}{s}=\frac{\kappa}{1}$. Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. Can an irrational number raised to an irrational power be rational? Can an irrational number raised to an irrational power be rational?  If it can be rational, how can one prove it? Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$. I want to learn Linux Kernel and I've downloaded the latest kernel version. Now I want to configure Linux with make menuconfig and I'm receiving an error:     Arch/x86/Makefile:184  Compiler lacks asm-goto support...   Can somebody help me?