Integrate $\int \frac{dx}{\sqrt{x^2-9}}$ by trig substitution
Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$
Non abelian Lie group with abelian Lie algebra Problem: Let $X$ and $Y$ be independent and suppose that each has a $\text{Uniform}(0,1)$ distribution. Let $Z = \min\{X, Y\}$. Find the density $f_Z(z)$ for $Z$. Hint: It might be easier to first find $\mathbb{P}(Z &gt; z)$.  Attempted Solution:  Given that $X, Y \sim \text{Uniform}(0,1)$, how do we not just have the following?  $$ f_Z(t) = f_X(t) = f_Y(t) = \begin{cases} 1 &amp; \text{if } 0 \le t \le 1 \\                                           0 &amp; \text{otherwise}                             \end{cases} $$  Of course I'm highly suspicious of this answer because it's not making use of the fact that $X$ and $Y$ are independent, nor is it making use of the provided hint. How to plot and solve the numerical solution of a integro-differential equation At the start of A New Hope, when the corvette is being attacked, C3PO says something to the effect of:     There'll be no escape for the princess this time.   But when Luke stumbles onto Leia's message in R2D2, C3PO seems to not know who the figure in the message is:     I'm afraid I'm not quite sure, sir. I think she was a passenger on our last voyage. A person of some importance, sir - I believe.   Is C3PO feigning ignorance to protect her, or does he really not recognize her? I got some new hard discs and would like to set my PC up in the following way:   Ubuntu is currently installed on one drive I want to put Windows on a second drive I would like a third drive for shared content (music and stuff) that both OSes can access     What I need to know is what format should the content drive be?  How should I configure the drive order in my bios for GRUB to be boot manager and how to configure GRUB for multi-HDD booting? how to avoid ssh asking permission? We are attempting to speed up the installation of oracle nodes for RAC installation. this requires that we get ssh installed and configured so that it doesn't prompt for a password.  The problem is: On first usage, we are prompted for  RSA key fingerprint is 96:a9:23:5c:cc:d1:0a:d4:70:22:93:e9:9e:1e:74:2f. Are you sure you want to continue connecting (yes/no)? yes   Is there a way to avoid that or are we doomed to connect at least once on every server from every server manually? Is there a method to export the URLs of the open tabs of a Firefox window? If I have a Firefox window open that contains 10 tabs, is there a way in Firefox or by a plug-in to get the URLs of those 10 tabs as a text file or some other format?  Right now if I want to do this I need to copy the URL of tab A, paste it somewhere, move to tab B, and repeat.  I could also bookmark all the tabs into a folder and export that, but that seems like such a hassle.  If there is no such method, could someone point me to some documents that describe the basics of writing a Firefox plug-in.  I am willing to write this myself if there is no "standard" way. Rewrite rules goes away Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path connected? Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path-connected?  I think it is, and I tried to prove it by contradiction, but I got stuck. Any hint helps! What is the difference between Shrink Database and File? DBCC ShrinkDatabase() DBCC ShrinkFile()    Do I need to run both DBCC commands in order to shrink the database? What is the difference between these two above? Can HIV be transmitted via mosquitos? It is known that HIV is usually transmitted by direct blood or body fluid contact between an infected individual and a healthy person (like blood transfusion or needle sharing):  Suppose a mosquito bites an individual suffering from AIDS and in the process sucks up some T cells infected with HIV along with RBCs. Then it bites another person not suffering from the disease, and transfers these infected T cells. Isn't there a high probability of the second individual contracting HIV? Can the determinant of a matrix can be made $0$? An entry of an $n \times n$ matrix with nonzero determinant is defined as interesting if by changing this entry (and only this entry) the determinant of the matrix can be made $0$.   Is it true that each entry of every matrix with nonzero determinant is interesting? Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? Remove application from Google Launcher Is the use of a hyphen between "non" and an adjective strictly necessary? Do I need to put a "-" between "non" and an adjective? As an example in physics we say "a non isolated photon", "non tight photon"... The context is very formal (paper publications and similar). Is there a general rule? Are there some differences between countries? Can I use Wubi for manual installation? Is it possible to use Wubi for Ubuntu 13.04? I read that it will not be supported in the official release due to some problems. Is this going to change in the future, and can we expect to see a future release with support for 13.04? I'm bugged by the following that's summarized on p. 109 of .     False theorem: All horses are the same color.      Proof by induction:      $\fbox{$P(n)$ is the statement: In every set of horses of size $n$, all $n$ horses are the same color.}$   $\fbox{Base Case or $P(1)$:}$ One horse is the same color as itself. This is true by inspection.   $\fbox{Induction Step:}$ Assume $P(k)$ for some $k \geq 1$.   $\fbox{Proof of $P(k + 1) :$}$   Since $\{H_1, H_2, ... , H_n\}$ is a set of $n$ horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.   Since $\{H_2, H_3, ... , H_{n+1}\}$ is also a set of $n$ horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.         Therefore, all $n +1$ horses in $\{H_1, H_2, H_3, ... , H_n , H_{n+1}\}$ are the same color. QED.   The issue the instructor was trying to point out is clearly valid. For the case $n = 1$, there is only ${H_1}$. So this case says nothing about possible overlapping elements of each set of $(n + 1)$, for instance $H_2$ in the above proof.    But it was proposed in the class discussion that this was the only problem. Had you been able to prove $P(2)$ true, then a proof of the above format would have been fine.    My interpretation is that yes, you could prove all horses are the same color, if you can prove that any set of two horses will be the same color. But this format would not work.  Why not? The problem I see is that the above proof is for the existence of at least one particular pair of sets of horses of sizes $n$ and $n + 1$, such that in each set, all horses are the same color.  Particularly when the set of size $n$ is a subset of the set of size $n + 1$.  In order to prove the induction step, don't you need to prove that sets of sizes $n$ and $n + 1$, do not necessarily contain the same, overlapping elements?  You could prove that any horse can be added to a set of 2 horses.  Take the last two, and they must be the same color, and so on.  Wwhatever color the first two happen to be, all other horses must thus be the same color.    Am I misinterpreting the example, or am I making a logical error? Thanks in advance. Layers with same coordinate system should align/overlap in ArcMap but do not? Is it safe to disable System Integrity Protection? While using OS X El Capitan 10.11.x, is it safe to disable System Integrity Protection? There are all these additional apps that modify folders and the dock that I want to install but you have to disable rootless first.  Good idea or bad idea? MPPT charging a li-ion battery I just designed this MPPT solar charge controller which outputs 4.2 V regulated via MPPT, it uses the SPV1040 IC. This will be used on a satellite with 4 side panels, each with 8 solar cells in parallel outputting 2.6 V at approximately 122 Ma. Each panel will have this exact circuit and all the outputs will be connected in parallel to a 3.7 V lithium ion camera battery.  When active only 1 solar panel will be active and the other MPPT circuits will be offline or providing a very low amount of current. How should I connect the 4.2v to the battery, just directly or using some sort of controller to stop charging at 4.2v? Keep in mind the battery will always have a load and it will charge for around 50m in a 90m cycle due to being eclipsed. I would obviously like the battery to last as long as possible so I am not sure what are the implications of constantly having a battery connected to 4.2 V during 50m. Thanks.   The schematic was made based off a simulation made using the ST Edesignsuite directly from the provider.  Excuse my previous question about this, wasn't properly worded. Intraclass correlation coefficient vs. F-test (one-way ANOVA)? I'm a little confused regarding the intraclass correlation coefficient and one-way ANOVA.  As I understand it, both tell you how similar observations within a group are, relative to observations in other groups.  Could someone explain this a little better, and perhaps explain the situation(s) in which each method is more advantageous?