Evaluate $$\int_0^1 \frac{\ln{\left(x+\sqrt{1-x^2}\right)}}{x} \; \mathrm{d}x$$ I tried $x=\sin{t}$ the symmetry with $t=\frac{\pi}{2}-t$ to get $$\int_0^{\frac{\pi}{2}} \frac{\ln{\left(\sin{t}+\cos{t}\right)}}{2\sin{t}\cos{t}} \; \mathrm{d}x=\int_0^{\frac{\pi}{2}} \frac{\ln{\left(\sin{t}+\cos{t}\right)}}{\sin{2t}} \; \mathrm{d}x$$ This looks much nicer I think but now what...? I considered $\sin{t}+\cos{t}=\sqrt{2} \sin{\left(x+\frac{\pi}{4}\right)}$ but that doesnt help.   Any hints or suggestions please. Integral equals about 0.61685028
By integration by parts and the substitution $x = \sin t$ we can easily calculate the integral $\int_{0}^{1} \ln (x+ \sqrt{1-x^2})dx$ which equals to $\sqrt{2} \ln (\sqrt{2} +1) -1.$ I’ve tried to use the same substitution $x = \sin t$ to calculate the integral $ \int_{0}^{1} \frac {\ln (x+ \sqrt{1-x^2})}{x}dx,$ which becomes $$ \int_{0}^{\frac {\pi}{2}} \frac {\ln \sin (t+ \frac {\pi}{4})}{\sin t}dt$$ It seems difficult to solve the particular integral. Any help?
Suppose that $X_1,X_2,...,X_n$ is a random sample from a normal distribution, $X_i\sim N(\mu,9)$.  Find the UMVUE (uniformly minimum variance unbiased estimator) of $P(X\le c)$ where $c$ is a known constant.  Do this by finding the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$ and apply the Rao-Blackwell theorem with $T=u(X_1)$, where $u(x_1)=1$ if $x\le c$ and zero otherwise.  So clearly the parameter that we're trying to estimate given the problem description can be written as $\Phi\left( \frac{c-\mu}{3} \right)$ where $\Phi$ represents the CDF of the standard normal distribution.  However, I get stuck when I need to try and find the conditional distribution.  Without knowing the joint distribution or the other conditional distribution, how can I possibly find the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$? Is a sentence always grammatically incorrect if it has no verb? Is the following grammatically correct? My friend says the second sentence is grammatically incorrect, but couldn't explain why.     I have always been fascinated by statistics. The different ways in   which you could look at data and infer knowledge from it. JavaScript open in a new window, not tab Extending my Root Paritition I've Win8/Ubuntu Dual boot. I already have 4 partition so i cannot create a new partition and mount it elsewhere. So, this () question did not help.  Here's a screen shot of my Gparted (run on Ubuntu Live)   How can I extend /dev/sda6  so that  i can make use of unallocated space ?  (w/o failing my current grub) Disable Narrator shortcut key in Windows I've install Windows 8 on a BootCamp macbook air The problem is that the command key located differently from the winkey in a normal keyboard, therefor, I keep clicking Winkey+Enter which starts the narrator (which is really annoying!)  Is there a way to disable this shortcut? 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At first I though it was: at the moment a king/lord dies, to find who holds the title next, you do a depth-first search down the family tree starting with the oldest sibling (ignoring anyone who isn't true-born or has given up their claim ie as Night's Watch or Kingsguard etc.).  Then, for Dornish law you take the first living person and for Westerosi law the first living male person found in this way (so succession can pass through a woman in both cases, but only Dornish law actually allows them to claim the title). If there are no valid living heirs, then you work backwards along the paternal line until you find an ancestor who has living heirs and start from them.  But there were various references to if Sansa's brothers were all dead, then her child would claim Winterfell.  It seems to me that that's only true if somebody else doesn't get it first.  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SCP through multiple hosts describes how to SSH into a machine that can only be accessed through one or more hosts (that can also be SSH'd into).  Is there an equivalent way of using scp? That is, copy a file to my local machine from the far machine (assuming I have password or key access to all machines along the way)? I have a string that represents a non indented XML that I would like to pretty-print. For example:  &lt;root&gt;&lt;node/&gt;&lt;/root&gt;   should become:  &lt;root&gt;   &lt;node/&gt; &lt;/root&gt;   Syntax highlighting is not a requirement. To tackle the problem I first transform the XML to add carriage returns and white spaces and then use a  tag to output the XML. 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Any ideas? need to move a domain controller, the first in the forest that holds all the fsmo roles, to another location in a clients building. it will require turning off this dc, call it dc1 for this question. i want to transfer the fsmo roles to a new domain controller call the new one dc2. dc2 is already on the network and has been promoted and made a domain controller, it's dns settings are set and it is also a global catalog (GC). the work is scheduled to be done after hours and i am also planning on moving the dhcp server to dc2.  i am looking for a best practices checklist of things to verify prior to moving fsmo roles and turning dc1 off? as far as i know there are no issues with replication between the dc's. my biggest worry is if i turn on dc1 after moving it and i have hardware issues or boot issues, i would rather move the fsmo roles to a known good one that is a few months old (dc2) vs still using a 5yr old box (dc1), this is part of my migration strategy too.   thanks for the help. 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