integrate from 0 to 1 $$\int \frac{x^7-1}{\log x} \, \mathrm{d}x$$![It's from MIT integration bee][1]  I try to solve this by substitution but nothing works out. Solving this problem from 10 days .I have no idea how they have a answer $\log(8)$.
/A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand?
How was the first log table put together? Henry Briggs compiled the first table of base-$10$ logarithms in 1617, with the help of John Napier.  My question is: how did he calculate these logarithms? How were logarithms calculated back then?  I've found   to be fairly useful, but they don't seem to say much in the way of what I'm looking for. Any answers or useful references would be appreciated. How to permanently set environmental variables I'm playing around with height maps (bitmaps), trying to create some of my own in my game, and for that I need to implement some basic drawing methods. I've quickly realized that drawing straight lines isn't as basic as I thought.  It simple's if your points share an X or Y coordinate, or if they are aligned so you can draw a perfectly diagonal line. But in all other cases its trickier.  What algorithm do you use to determine what pixels need to be coloured for it to become a "straight" line ? I recently moved into an in-law unit of a large old house.  In the kitchen, there are two pipes coming out of the wall (one protruding out a little more than the other).  I had assumed they were gas lines to connect my stove.  But since the stove only connects to one pipe, I'm now wondering if they're actually water lines. Can anyone explain how I can tell the difference? On the other side of the wall is a large closet with the air conditioning system, water heater, and what I think is the gas shutoff valve. Showing that the roots of a polynomial with descending positive coefficients lie in the unit disc. Reputation changes not visible in the achievement box (again) Just like , which was marked as . But today, it's not working again. I'm still notified of new badges:      Update: it looks like new events are now updated; reputation events from earlier this UTC afternoon are still missing: Single quotes vs. double quotes in Python According to the documentation, they're pretty much interchangeable.  Is there a stylistic reason to use one over the other? Prologue:  This is an effort to follow up on  which is marked as a duplicate of  highly voted and status-declined feature request (although they seem to be different, one focusing on the OP and the other on the downvoter).   My New Year's resolution was to downvote more.  So on January 3rd or so, I saw a post that was low quality, and downvoted.  Two days later, I came back to see if it was ready to be deleted or had gotten piled with downvotes, to find it with a significant positive score!  At which point, I noticed that the answer had been significantly improved.  So I retracted my downvote, changed it to an upvote, and immediately resumed my dearth of downvoting again.  I think I have a responsibility to downvote bad posts, despite that oh-so-painful -1 rep penalty.  But if I am also going to have the responsibility to follow up on every downvote I cast, I'm not going to bother. I don't want my downvote to leave a residual penalty on posts that were once low quality, and I am not going to trouble myself to keep checking up on every post I downvoted.  Can a downvoter have an option to be notified when a post they voted down was edited?  I wouldn't make this an automatic thing, because not all downvoters want it, and it isn't applicable to all cases of downvoting (although  that this should be part of the responsibility of a downvoter).  Personally, I would make this a universal toggle on-off option on the user profile page, not a question-by-question option, but I don't have strong feelings about it. How can i set different wallpapers for Ubuntu 15.04 login and Desktop This has been bothering me very much. i have memory that Ubuntu 12.04 had this configuration? am i wrong? finally, is it possible for the login screen to have different wallpaper from the desktop? is this possible in Ubuntu 15.04? i would appreciate a stable understandable response. thanks :)  -ben Had it not been vs If it were not for How to safely replace a not-yet-failed disk in a Linux RAID5 array? Prevent people from downloading / copying / making a copy of data in Google doc High-end compact camera with swivel screen and running on AA batteries I'm looking for a new compact camera that:   has a  is small enough to be mountable on a  and, runs on AA batteries. good flexibility   Requirement (3) stems from the fact that I may be hiking in the wild for weeks on an end, with no opportunity to change a Lithium-Ion battery. If another solution permits me to take thousands rather than hundreds photographs on full charge, I may consider that as well.  Additionally, it would be nice (but not essential) to have:   the ability to handle high dynamic range a viewfinder physical buttons — I live in a cold climate, and don't like to use touch-screens   The upper price range for compact models is still good for me, and I don't mind a pricerange of €400–€800 (but I don't mind cheaper either, if still of good quality).  I've been the happy user of several models in the , but now my PowerShot A650 IS has died. Starting from , it seems requirement (3) kills of many cameras: the , (might also fail (2)), the , the , the ... Although enough models meet requirements (1), (2), and (4), as well as the (non-essential) additional features I listed, it seems in the past 5 years, cameras have moved away from using AA batteries. What camera models still exist that meet my requirements?    Last photograph taken with my Canon PowerShot A650 IS, taking on Christmas Day 2012 in the Caldera de Taburiente, La Palma, Canary Islands, Spain. I have asked this question on SO but perhaps that was the wrong location. Apologies for double posting but the question may have a better audience here.  The original question is:   And the text is below. Thanks.    When running the postgresql [9.3] "COPY TO" functionality, the postgres user creates a file. I have told it to put the file in the tmp directory (although this can be changed if needed).  This is run as part of a cron job, the user is an "export" user set up especially for this task. The end goal is for the exported file to end up in /home/export/created_file.csv.  1) postgres can't stash the exported file straight into the home directory of another user without setting up the folder to be overly permissive.  2) the export user can copy the file from /tmp to it's home directory, but then can not clean up (delete) the postgres owned file in /tmp.  This is a daily process and so old files will need to be deleted in some way. I can add a cron job to clean up the files afterwards but I was hoping for a cleaner solution - either a way to change the permissions on the file that postgres creates, or some way to give the export user the additional permission of being able to delete that file in advance of it's creation.  Giving the export user the same permissions as postgres in general however isn't an option, as it would then have free reign over the database.  Any advice, be it unix based or postgres based would be very welcome, thanks. Characterization convex function. Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that for all $a,b\in [0,1]$ with $a&lt;b$ $$f\left(\frac{a+b}{2}\right)\leq\frac{1}{b-a}\int_a^b f(x)\,dx.$$ How to prove that $f$ is a convex function? Thank you for your help. Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$? Recently I came across a debate about whether it's fair to call RAW 'unprocessed' or 'unaltered' sensor data of a DSLR. As far as I know, the analog sensor data is processed into a digital RAW file using the ISO setting, a process during which the Bayer-sensor data is demosaiced as well. However, those are just the necessary steps for saving a digital file while maintaining maximum quality.  Are those assumptions correct? If so, are there any additional steps that need to be performed between the sensor receiving light and the saving of the RAW file? And can those be reasonably be described as an 'alteration' of the sensor data, or is it fair to say that a RAW file holds the unaltered sensor data? Is it possible to exclude questions I've already answered from the search criteria? What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime? Benevolent aliens who (initially) refuse to show themselves This is a sci-fi book where aliens arrive on Earth and push Earth towards world peace, but refuse to reveal themselves physically. When the aliens finally do show themselves, we find that they       look like .   I recall that the aliens seemed benevolent, but that humanity did not like the direction they were being told to go. I cannot remember the ending or if the aliens turned out to be truly benevolent.