I got this challenge question on brilliant.org: $\int_{0}^{\pi/4}\sqrt{\tan x} dx$  I first used, $\int_{0}^{\pi/4}\sqrt{\tan(\pi/4-x)} dx$ which simplifies: $\int_{0}^{\pi/4}\sqrt{\frac{1-\tan x}{1+\tan x}} dx$  Then, I used Weierstrass's substitution by using $t=\tan(\frac{x}{2})$ and $dx=\frac{2}{1+t^2}$ and simplified: $2\int_{0}^{\tan(\pi/8)}\frac{1-t}{(1+t)(1+t^2)} dx$  After some manipulation, I calculated the integral to be $I=2\ln(\tan(\frac{x}{2}))-\ln(1+t^2)+c$ and $I(\pi/4)-I(0)=0.53$  But Desmos tells something different:  Have I done something wrong? I have rechecked a few times. But I find no error!
I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$
Label PGFPlot table points with coordinates I would like to label each plotted point with its x and y coordinate. I am able to label with a single one using the MWE below. I must avoid manually labeling each point since my actual table size is larger and will be changed out regularly. \documentclass{standalone}   \usepackage{tikz} \usepackage{pgfplots} \pgfplotsset{compat=1.16}  \begin{document}  \pgfplotstableread[col sep=&amp;,row sep=\\]{  x     &amp;  y     \\  0.940 &amp;  0.992 \\  0.992 &amp; -0.940 \\ -0.992 &amp;  0.940 \\ -0.940 &amp; -0.992 \\ }\mytable  \begin{tikzpicture}  \begin{axis}     \addplot+[only marks, nodes near coords]table[x=x,y=y,meta=x]{\mytable}; \end{axis}  \end{tikzpicture} \end{document}  which creates the following figure  The desired output is for each point to be labelled with both coordinates. For specific example the top left point would be labelled with &quot;(0.94, 0.992)&quot;. It’s a story of a female protagonist who has a harsh life. Just as she is about to die, she time travels back into her young self. To avoid the future she already lived, she decides to gets strong by learning swordsmanship. After a while, she joins the army under the command of a prince who wears an eye patch. She is quite OP, and helps the prince. It was kind of a medieval setting manga, with dukes and earls and count types. Time is when the use of sword and magic is prominent. The new GNU Octave 4.0.1 was released March 23, 2016.  I would like to find some standard way to install it in El Capitan (mine OS X 10.11.3) because it has significantly better support in some functions than Matlab, discussed in the thread .  I do brew search octave and get  homebrew/science/octave    Caskroom/cask/octave       Caskroom/cask/xoctave       where one of the first two should be a correct one. The project's homepage  for OS X is down  about installing Octave on OS X systems.  There is also ruby's way of installing Octave described in the thread  but for Yosemite.  I would like to find some standard way of getting Octave for OS X.   I do successfully brew update and brew upgrade before the commands. My Homebrew is 0.9.5 (git revision 80d1; last commit 2016-03-24).  . I run brew info octave but I get  Error: No available formula with the name "octave"     How can you install Octave in some standard way for El Capitan? Is a switches voltage and current rating a hard limit? Capturing console output from a .NET application (C#) How do I invoke a console application from my .NET application and capture all the output generated in the console?   (Remember, I don't want to save the information first in a file and then relist as I would love to receive it as live.) PHP code is not being executed, instead code shows on the page Is it possible to detect a player consuming food items? Seasonal Loan Amortization Where To start Learning With Salesforce Development Where should I start learning? There are various topics to learn like Objects, Classes, Apex, Visualforce, SOSL, SOQL, Triggers, etc. There are so many options out there to start learning. I couldn't decide what topic to learn first, and then step by step get proficiency in all aspects of Saleforce. For $n \geq 2$, prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$ For $n \geq 2$ prove that $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  We need to use induction.   The Principle of Mathematical Induction, Theorem 4.2.1, states that $n_0 \in \mathbb{Z}$. For each integer $n \geq n_o$, let $P(n)$ be a statement about $n$. Suppose that the following two statements are true:   $P(n_o)$ $( \forall n \geq n_o)[P(n) \rightarrow P(n+1)]$   Then, for all integers $n \geq n_o$, that statement $P(n)$ is true.  Suppose,  $(1- \frac{1}{4})(1- \frac{1}{9})(1- \frac{1}{16})...(1- \frac{1}{n^2}) = \frac{n+1}{2n}$, the equation, is true for some natural number $k$.  $\frac{1}{2^2})*...*(1- \frac{1}{k^2}) = \frac{k+1}{2k}$  The induction hypothesis $P(k) \rightarrow P(k+1)$ must hold for any natural number $k, P(k),$ from the basis and inductive step.  $(\frac{1}{2^2})*...*(1- \frac{1}{k^2})*(1- \frac{1}{(k+1)^2}) = \frac{k+1}{2k}(1- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $\frac{k+1}{2k}(\frac{(k+1)^2}{(k+1)^2}- \frac{1}{(k+1)^2})$  $\leftrightarrow$ $(\frac{(k+1)}{2k})( \frac{(k+1)^2-1}{(k+1)^2})$  $\leftrightarrow$ $( \frac{(k^2+2k+1-1}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{(k^2+2k}{(2k)(k+1)})$  $\leftrightarrow$ $( \frac{k+2}{(2)(k+1)})$  We need to demonstrate that   $(\frac{1}{2^2})*...*(1- \frac{1}{n^2}) = \frac{n+1}{2n}$  holds true for $n \geq 2$  The question is do I just substitute a number that is greater than two after I did the induction? Kendo Grid & Rich text Editor of Sharepoint 2013 How does one get started with web test automation using Selenium? So I've been testing web based analytical software now for the past 4 years or so with mostly manual techniques. But for 2015, I've set myself the goal of getting setup and start writing automated tests and to introduce automated testing into my company.  I've done a little research and it seems that Selenium is quite big when it comes to writing automated tests, but most of the tutorials I've come across reference the Java language.  Some questions:   Is Java the primary language used for Selenium tests?  Do you recommend it over C# ?  Wat is the process of writing test scripts?  Are there Frameworks that I should use that can make the process even easier? Find the limit of $\frac{n^{x}}{(1 + x)^{n}}$ Let $x&gt;0$. Find $$\lim_{n\to\infty}\frac{n^{x}}{(1 + x)^{n}}.$$   What I was thinking involved an inequality, by reason of considering the fact that  $$(1 + x)^{n} &gt; C(n, k) x^{k}$$ for some $k$ such that $n&gt;k$. Expanding the binomial coefficient and taking the $n$ common from each bracket, we have something like $$(1 + x)^{n} &gt; C(n, k) x^{n} &gt; \frac {x^{k}n^{k}}{k! 2^{k}}.$$ But now, I am totally clueless. By intuition, I know that the limit has to be zero, so I just want to prove using the concept of inequalities that what I thought was the right approach (being the sandwich theorem). Any help would be appreciated! Evaluating $ \lim_{x \rightarrow 0}\left (\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$ Escape velocity from a rotating black hole Under Newton, the escape velocity is  $$v_{esc} = \rm c \ \sqrt{r_s/r}$$  where $\rm r_s=2 \ GM/c^2$. In the nonrotating relativistic case (the Schwarzschild case) the radial escape velocity is the same:  $$v^{\parallel}_{esc} = \rm c \ \sqrt{r_s/r}$$  which is the speed of light, $v={\rm c}$, at the event horizon at $\rm r=r_s$. Since the photon sphere, where an object with $v={\rm c}$ would not escape, but be captured into orbit, is at $r=3 \ {\rm GM/c^2}$, the transverse escape velocity is  $$v^{\perp}_{esc} = \rm c \ \sqrt{r_s/r}\div \sqrt{1-r_s/r}$$  and for the observed (shapiro delayed) velocity ${\rm v}$ the transformation is  $${\rm v}^{\perp}=v^{\perp} \ \sqrt{\rm 1-r_s/r} \ , \ \ {\rm v}^{\parallel}=v^{\parallel} \ (\rm 1-r_s/r)$$  so the observed angular motion is the same as with Newton, yet locally higher, while the observed radial motion is slowed down because of the radial length contraction and time dilation.  But what about the Kerr case where the rotation is significant and the event horizon is at $\rm r=(1+\sqrt{1-a^2}) \ {\rm GM/c^2}$ instead of $\rm r=r_s$?  A testparticle which is locally at rest and therefore corotating with the angular velocity of the frame dragging  $${\Omega = \rm d \phi/ {\rm d t} = {\rm 2 \ a \ r/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  should experience the time dilation factor  $$\rm \dot t = {\rm d t}/ {\rm d}\tau = \sqrt{{\rm ((a^2+(r-2) \ r) (a^2 \cos ^2(\theta )+r^2))/((a^2+r^2)^2-a^2 (a^2+(r-2) \ r) \sin ^2(\theta ))}}$$  relatively to the far away observer, so multiplying $\Omega$ with $\rm \dot t$ should give the local frame dragging angular velocity, which, multiplied with the $\rm \sqrt{x^2+y^2}$ value of the Boyer-Lindquist to cartesian transformation should give the transverse velocity - this is slightly below $\rm c$ for a maximaly rotating black hole (with $\rm a = 1$ - at the edge of the outer equatorial ergosphere where $\rm r=1$ and $\rm R=\sqrt{r^2+a^2}$).  But how do I find the radial and total escape velocity, and what are the rules to convert the shapiro-delayed coordinate velocities $\rm d r/d t \ , \ d \phi/d t \ , d \theta/d t$ to the local 3-velocity components $v^{\parallel}$ and $v^{\perp}$ of a test particle (local meaning relatively to a frame dragged probe with contant $\rm r$)? The radial length contraction seems a bit different than in the Schwarzschild-case, where it is the same as the gravitational time dilation factor... If I have a Windows executable, how can I find out which dlls it will load?  I'm just talking about which ones that will be loaded statically, not ones it might load dynamically with something like LoadLibrary. Divisibility, LCM and GCD for fractions (rationals) how do i find lcm of two fractions?  For example:  $\frac{2}{3}$ and $\frac{5}{8}$ To env or not to env Are there any examples of Humans and Aliens producing offspring in Star Wars? The title is pretty self explanatory, although to expand on the question further I am looking for any examples of inter-species breeding in the Star Wars universe, not just inter-species (ahem) "relations". $P(X<Y)$ where X and Y are exponential with means $2$ and $1$