How can we calculate this integral:     $$I=\int\limits_0^1\frac{x^n-1}{\ln(x)}\,\mathrm dx$$   I believe that integral is equal to $\ln(n+1)$, but I don't lnow how to prove it.
$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$  As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
How to reshape data from long to wide format Scale a div to fit in window but preserve aspect ratio The pdf of $\frac{X_1-\bar{X}}{S}$ Suppose $X_1, X_2,...,X_n$ be i.i.d from $N(\mu,\sigma^2)$ with unknown $\mu \in \mathcal R$ and $\sigma^2&gt;0$  Let $Z=\frac{X_1-\bar{X}}{S},$ S is the standard deviation here.  It is can be shown that  $Z$ has the Lebesgue p.d.f.  $$f(z)=\frac{\sqrt{n} \Gamma\left(\frac{n-1}{2}\right)}{\sqrt{\pi}(n-1)\Gamma\left(\frac{n-2}{2}\right)}\left[1-\frac{nz^2}{(n-1)^2}\right]^{n/2-2}I_{(0,(n-1)/\sqrt{n})}(|Z|)$$  My question is then how to get this pdf?  The question is from  in example 3.3.4  to find the UMVUE of $P(X_1 \le c)$. I can understand the logic and procedures to find the UMVUE but don't know how to get the pdf.  I think this question also relate to this    Thank you very much for help or point to any related references will be also appropriated. Changing local IP address with networksetup command on OSX? My computer routinely receives connection from another host in the local network. However, the router changes its ip address quite frequently, and maybe netsh's equivalent, networksetup, can help? A sequence includes $a_p=\sqrt2$, $a_q=\sqrt3$, $a_r=\sqrt5$ for some $1\leq p<q<r$. Can these be terms of an arithmetic progression? harmonic? An analytic function $f$ on $\{z \in\mathbb{C} : |z|>1 \}$ If "I'm thinking" is correct, why isn't "I'm knowing" correct in formal English?  think+ing = thinking is correct know+ing  = knowing isn't correct   Why? If you are reading this, you may have been affected by one of our post quality bans. If so, it is important that you  to understand what has happened and how to regain the permissions that you lost. While trying to ask a question, one could see:  We are no longer accepting questions from this account. See  to learn more.  Likewise, for answers:  We are no longer accepting answers from this account. See  to learn more.   Why am I getting this message? Are deleted posts taken into account too? Is a question/answer ban the same as a suspension? How do I avoid getting a question ban? How long do I have to wait before I can post again? What can I do to release the ban? How can I reactivate my account? Can I simply create a new account? I'll just ask somewhere else on the SE network, and they'll migrate my question to the correct site! Will a ban on one Stack Exchange site affect my standing on other sites in the network? Does this apply to meta sites too? My account is in good standing. Why am I still blocked? So I defeated a Team Valor gym earlier today using my highest cp Pidgeot. It turned grey, so I went to claim it with that Pidgeot, except it wouldn't let me. What determines which pokemon you are allowed to place at a gym? Is it just those who have participated in a battle can't? Is there a time period for it? What about training at a friendly gym? Does this impact anything? Using MySQL, I can do something like:  SELECT hobbies FROM peoples_hobbies WHERE person_id = 5;   My Output:  shopping fishing coding   but instead I just want 1 row, 1 col:  Expected Output:  shopping, fishing, coding   The reason is that I'm selecting multiple values from multiple tables, and after all the joins I've got a lot more rows than I'd like.  I've looked for a function on  and it doesn't look like the CONCAT or CONCAT_WS functions accept result sets.  So does anyone here know how to do this? How to change the date format? Presently the date is showing as mmm dd hh:mm:ss. Is there any way that I can change it to as per my liking. For example : dd mmm, hh:mm:ss. Correct usage of will and would Solving ODE with negative expansion power series I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:  $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$  $g_{ii}$ are given, from metric, but that's not important. I need to assume that the solution (since I'm looking for components of $\xi^\mu$, which is a vector with components $\xi^t,\xi^r,\xi^\phi$) is given with power series of the form:  $\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, and this is to be seen as expansion around $1/r$ (expansion around $r=\infty$).  Now when I plug this in the ODE I get this  $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where   $\xi^\mu_{n,i}$   is the derivative with the respect to i-th component.   What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my $\mathcal{O}(r)$ terms cancel each other out? Or?  I'm kinda stuck :\ Is there a package that supports the addition or subtraction of chemical equations, perhaps with automatic tabbing and a alignment.  Whilst it is obvious I could set up tables or use mathmode's align feature, it seems so much more efficient to have a package handle the tabbing automatically, since it is rather predictable in this application.  mhchem does not seem to have such a feature...      Another advantage is you could maintain the readability of your equations and you would not have the added clutter associated with tables, especially tabs &amp;, which diminish the readability of your .tex document.  And of course, the line should be automatically inserted before the last row. I really read many tutorials, but crontab doesn't start notify send.  What I've tried:   etc/crontab  added at the end:  15 * * * * notify-send 'crontab' 'test'  with crontab interface:  crontab -e   added at the end:  15 * * * * notify-send 'crontab' 'test' Possible Duplicate:          I want to know some example of a continuous function which is continuous at exactly one point. We know that $f(x)=\frac{1}{x}$ is continuous everywhere except at $x=0$. But i think this in reverse manner but i dont get any example. So please help me out! Determine Whether Two Date Ranges Overlap Given two date ranges, what is the simplest or most efficient way to determine whether the two date ranges overlap?  As an example, suppose we have ranges denoted by DateTime variables StartDate1 to EndDate1 and StartDate2 to EndDate2. What is the difference between independent and mutually exclusive events? Movie Identification: People record their experiences on reel-to-reel I know that this sounds like Strange Days, which I haven't seen, but from the Wikipedia description I don't think that's it.  I saw this back in the early-to-mid-90's on TV, pretty sure it was a premium channel. It follows a research group who develops a way to record experiences. Not thoughts, but at least the sights, sounds, smells, touch, etc. They record it onto this reel-to-reel tape, it looked striated and golden in color. The protagonist is the head of the research team.  I don't remember a great deal of the plot, but there were conspiracies surrounding the invention. The lab clown decides to give it a whirl by putting the bike-helmet-sized apparatus on his head and going out to have a little fun, in the classic wine, women, and song fashion. Once he returns, an older member of the team finds the tape and finds a section where the clown is having sex. Later on, the protagonist, who is the lead researcher, finds the older member shut up in his house seemingly for days and nearly comatose. The older researcher had the playback device over his head and is non-responsive. Worried, the protagonist removes the helmet and puts it on himself, only to find that the older researcher had snipped a few seconds of tape and had them running on a non-stop loop; the moment of orgasm for the clownish researcher. The older researcher eventually comes to and awkwardly apologizes.  I think there were also experiments in recording the sensorum of a lab monkey, and that played into the plot. I remember the plot seemed fairly thin and dull for what seemed to be a pretty neat idea. I don't desperately feel the need to watch the movie, but I'm mildly curious if my impressions would differ 20 years later now. I want to appropriate line segments that share the same geometry with a particular line width and labels based on the aggregated values of one field "GIS@BUS_5_Min".  My attribute table and polyline objects look as following, see image below   "BUSLINIE" means a bus line, "Overlap" means if the line has overlaps, i.e. it's geometry repeats.  Values of the field "GIS@BUS_5_Min" for these three objects has to be aggregated as one value and represented as a label sum("GIS@BUS_5_Min").  And the width of this line (selected in yellow) will be sum("GIS@BUS_5_Min")/100 = 19.10/100 = 0.1910  How can I achieve such output?  I assume it is possible with Layer Properties &gt; Width &gt; Data defined override &gt; Edit..., but I do not know how to check the same geometry and how to sum the values of certain features with CASE WHEN condition THEN result END conditional function?  Dissolve function does not work in my case because I do not have any key-fields.    References: