$$\int \dfrac{1}{1+\sqrt{\tan x}}\quad dx$$ $$\int \dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\quad dx$$ It is difficult for me to solve this integration.
We have to evaluate the following integral:  $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$  I tried this:  I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$.  But after that I got stuck.  The book where this is taken from gives the following as the answer: $$\ln(1+t)-\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$
Assume $0 &lt; a_0 \leq a_1 \leq \cdots \leq a_n$, show that equation   $$ p_n(z) = a_0 z^n + a_1 z^{n-1} + \cdots + a_{n-1} z + a_n = 0 $$   does not have root in $|z| &lt; 1$. Here $z$ is complex.   I started by considering the case in which $n = 2$, and tried to use the triangle inequality, while no progress was made. Any hints will be appreciated. What is mathematical existence? I am pretty sure $(B_{t}^{2})$ not Markov because the squared random walk is not.    I guess I can repeat the method since to be Markov it must satisfy the discrete.  Thanks The ls -al command shows the following output;  -rwxrw-r--    10    root   root 2048    Jan 13 07:11 afile.exe   What are all the fields in the preceding display? I have two question concerning information content of qubit.  Question 1: How many classical bits are needed to represent a qubit:  A qubit can be represented by a vector $q = \begin{pmatrix}\alpha \\\beta \end{pmatrix}, ~~ \alpha, \beta \in \mathbb{C}$. So, we need four real numbers to represent it. But when facts that (i) $|\alpha|^2+|\beta|^2 = 1$ and (ii) two qubits which differ in global phase only are indistinguishable, are taken into account, only two real numbers are necessary (angles on Bloch sphere). Moreover, we can choose a precission of the qubit representation (i.e. number of decimal places in $\alpha$ and $\beta$ or Bloch sphere angles) which influence number of classical bits needed to described the qubit.   So, it seems to me that the qubit representation cannot be used for measuring qubit information content but only memory consumption in simulation. Am I right?  Question 2: What is an (effective) information content of qubit:  A superdense coding enable us to send two classical bits via one qubit. But on the other hand you need two entangled qubits prepared in advance.  Given these facts, what is an information content of qubit? One or two classical bits? Or do I need to use another angle of view given the fact that qubits are "continuous" (i.e. described by complex numbers $\alpha$ and $\beta$)? Bold name in bib when having a prefix in last name Are players allowed to pass a penalty kick? How can I make sharp geometric 3d models look more realistic? I'm making a scene in Blender with manufactured mechanical and electrical objects - cylinders, metal objects and so forth. I want to make these look real, be fully present with a sense of mass and substance.  3DCG has a tendency to make everything very sharply geometric, often looking mathematical and artificial. In other scenes, some organic curves, normal maps or particle systems would help make object look real. Like for example, making a simple plastic dog model look more believable, one can add more vertices for better definition, add texture layer, add fur etc.  My scene shows clean resistors, wires, metal strips and such, all sharply geometric. It's tricky to find ways to make the result less mathematical and more substantial, realistic and interesting to the eye.  What are some effective ways to add realism to these objects? How to evaluate $\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$? For $\alpha \in \mathbb{R}$, define $\displaystyle I(\alpha):=\int_{0}^{2\pi}e^{\alpha \cos \theta}\cos(\alpha \sin \theta)\; d\theta$. Calculate $I(0)$. Hence evaluate $\displaystyle\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta)\; d\theta$.  Hint: To evaluate the integral that expresses $\displaystyle\frac{dI}{d\alpha}$, consider $\displaystyle\frac{\partial}{\partial \theta}(e^{\alpha \cos \theta}\sin(\alpha \sin \theta))$.  How do I do this question? I think this might have something to do with the Fundamental Theorem of Calculus, but I'm not sure.  I computed $\displaystyle I(0)=\int_{0}^{2\pi} d\theta=2 \pi$, and $\displaystyle I(1)=\int_{0}^{2\pi}e^{\cos \theta}\cos( \sin \theta) d\theta$. Following the hint I get  $$\begin{align} \frac{\partial}{\partial \theta}(e^{\alpha \cos \theta}\sin(\alpha \sin \theta)) &amp; =\alpha e^{\alpha \cos \theta} \sin (\alpha \sin \theta) + e^{\alpha \cos \theta}\cos(\alpha \sin \theta) \alpha \cos \theta \\ &amp; =  \alpha e^{\alpha \cos \theta} \sin (\alpha \sin \theta) + \frac{dI}{d \alpha} \cos \theta. \\ \end{align}$$  Is this correct so far?  The answers in the question referred as a duplicate does not help. I'm in a course dealing with real values, not complex. I am asked to solve the following question using central limit theorem. In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.  It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem. Been doing tests in sandbox in which I've created about a million accountshares that I need to get rid of for testing.   I'm trying data loader in bulk api mode to export than delete but I keep getting:     InvalidBatch : Failed to process query: QUERY_TIMEOUT: Your query request was running for too long. Trying again later.;   Not sure if I should be waiting longer? Any other options or dataloader configurations I should be looking at?  EDIT  I've been trying with batch apex jobs yet still having issues.   Script  string str='select id from AccountShare where lastmodifiedbyid =\'0050P000007Kfy2\' limit 10000'; sharedeletebatch bdt=new sharedeletebatch(str); Database.executeBatch(bdt); system.debug(bdt);   Batch  global class sharedeletebatch Implements Database.batchable&lt;sobject&gt; {       global final string query;      global sharedeletebatch(string q){            query=q;      }       global Database.QueryLocator start(Database.BatchableContext BC){        return Database.getQueryLocator(query);      }      global  void execute(Database.BatchableContext BC,List&lt;SObject&gt; scope){          delete scope;      }     global void finish(Database.BatchableContext BC){     }   }      First error: [REQUEST_RUNNING_TOO_LONG] Your request was running for too long, and has been stopped. Rules for refreezing food Keytool Signing Problem: Keystore was tampered with, or password was incorrect What is the difference between Θ(n) and O(n)? If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval? This is from Exercise 7 in p. 92 in Munkres's Topology.  Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray.  But I cannot start my proof because I cannot use $\sup$ or $\max$ functions.  What should I do? How can I recover deleted photos from an SD Card? My technophobe mother returned from a European trip to discover she'd erased all but 15 images from a nearly full 2GB card. As I am the computer guy in the family, she posted the card to me and asked that I make something magical happen.  The photos were taken with and deleted by a Nikon Coolpix. (I don't know the model.)  I am a Linux guy, but there is a Windows Vista box at my disposal.  What is my best strategy to try to recover something? What is the purpose of delayed suspend (Ctrl-Y) in Bash? $(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$ Finding rank-nullity of a given matices After upgrading to Ubuntu 14.04 from 12.04 I have duplicate entries in dash.    How can I get rid of them?