I got this problem from my textbook:  $ \sin{\theta} \mapsto \frac{e^{i\theta}-e^{-i\theta}}{2i} = \frac{z - \bar{z}}{2i} $ and $ z := e^{i\theta}$. So $ \frac{dz}{d\theta} = i e^{i\theta} \rightarrow d\theta = -ie^{-i\theta}dz= -i\bar{z}dz.$ What are the limits? Pluggin in $ \theta = 0 \rightarrow z = 1$ which is the same as $ \theta = 2 \pi $?
$$\int_0^{2\pi} \frac{1}{2+\sin(x)} \ dx$$  $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$  $$z=\alpha(x)=e^{ix}$$   $$\sin(x)=\frac{z^2-1}{2zi}$$ $$2+\sin(x)=\frac{4zi+z^2-1}{2zi}$$ $$\frac{1}{2+\sin(x)}=\frac{2zi}{4zi+z^2-1}$$    $$\int_\alpha \frac{2zi}{4zi+z^2-1} \ \ \frac{1}{zi}$$    $$4zi+z^2-1=0$$ $$z_1=\frac{-4i+2\sqrt{3}i}{2}$$ $$z_2=\frac{-4i-2\sqrt{3}i}{2}$$    $$\rvert z_1 \rvert &lt;1$$ $$\rvert z_2 \rvert &gt;1$$   Using Residue theorem:   $$\int_\alpha \frac{2}{4zi+z^2-1}=2 \pi i \ \lim_{z\rightarrow-2i+\sqrt{3}i} \ \frac{2}{z+2i+\sqrt{3}i}=\frac{2 \pi}{\sqrt{3}}$$    Is it correct?  Thanks!
Boot drops to a (initramfs) prompts/busybox I am running an HP pavilion dv6000 dual boot win7 and Ubuntu 12.04. (well, up until today).  After a reboot, the boot process drops to the BusyBox shell and I end up at the prompt:  BusyBox v1.18.5 (Ubuntu 1:1.18.5-1ubuntu4) built-in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs)   I've been researching others who have had this same problem, but haven't been able to find any of those solutions to work for me.  I tried the method described , and after the final command   mount -t ntfs-3g /dev/sda1 /root -o force   it does nothing and gives me another (initramfs) prompt.  I can boot to a live CD (USB) and get to a terminal, but it doesn't seem to do much good, as I can see the /dev/sda1 in the ls command, but it doesn't recognize it when I try to cd to it.  One more question: using the command fdisk -l how can I tell which mount point (sda1/sda2) is my windows partition and which one is Ubuntu? Nested roots sequence, how to prove it's monotone and bounded? Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one! No CUDA and GPU option $a\mid b,\ c\mid d\,\Rightarrow\ ac\mid bd $ $\ \, \bf\small [Divisibility\ Product\ Rule]$ I just need to check the reasoning in my proof is correct, I think it is valid although I'm not totally convinced because I can't follow the logic; does proving that $x$ is an integer prove that $ac|bd$?  Theorem: Let $a$, $b$, $c$, $d$ be integers. If $a|b$ and $c|d$, then $ac|bd$.  Proof: $aj=b$ and $ck=d$ for integers $j$, $k$. Then, $ac|bd$ implies $acx=ajck$ and thus $x=jk$ for some $x$. Since the product $jk$ is an integer, $x$ is an integer and thus $acx=ajck$ and thus $ac|bd$. What is the translation of what's written below?  I think it may be the poem of the One Ring, but I'm not sure. Which LaTeX to install on Linux? Which LaTeX distribution would you recommend for using on Linux from command line? I am looking for one with most complete coverage of packages. Can a website know about my browsing history? When I visit a website it sometimes puts cookies on my browser. Usually cookies keep track of browsing info from that particular site. If I don't delete those cookies and browse other websites, will the cookies be able to know which site I am browsing? Also, can any website read browsing history which is saved by browsers? Can any website know about my browsing history by help of cookies, browsing histories saved by browsers, or by any scripting language e.g. javascript? why this code can run ? double a[3]; a[1,1]=1; Is there a way to show statistic info? I just came across this idiom in some open-source Python, and I choked on my drink.  Rather than:  if isUp:     return "Up" else:     return "Down"   or even:  return "Up" if isUp else "Down"   the code read:  return isUp and "Up" or "Down"   I can see this is the same result, but is this a typical idiom in Python? If so, is it some performance hack that runs fast? Or is it just a once-off that needs a code review? Are proofs by contradiction really logical? Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction.   My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?  That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true". What is the difference between IQueryable and IEnumerable? What is the difference between IQueryable&lt;T&gt; and IEnumerable&lt;T&gt;?    See also  that overlaps with this question. Parametric vs non-parametric machine learning methods I looked-up many references and websites and researched on how to determine if a method is between parametric or non-parametric. I came up with below definitions,     A parametric algorithm has a fixed number of parameters. In contrast, a non-parametric algorithm uses a flexible number of parameters, and the number of parameters often grows as it learns from more data.   From .  Moreover, I found,     A parametric model, we have a finite number of parameters, and in nonparametric models, the number of parameters is (potentially) infinite.    From .  And many other methods of determination, though the problem is none of them can helo ones to determine if a certain hypothetical method is parametric or not. (For instance, why k-means' number of parameters are constant but KNN is variable or basically what do we call a parameter and what we do not?) Implicit Differentiation Help I have to use implicit differentiation to find $\frac{dy}{dx}$ given:  $$x^2 \cos(y) + \sin(2y) = xy$$  I don't even know where to begin, I missed the class where we went over implicit differentiation, and because of that, I am completely stuck.   Thank you everyone.  Edit: I don't know how to make the equation look all nice and whatnot, so sorry about that If I'm given a complex number (say $9 + 4i$), how do I calculate its square root? "She will be visiting" vs. "she will visit" When she goes to Mexico, she will be visiting Chichen Itza. When she goes to Mexico, she will visit Chichen Itza.  Is there a particular difference in meaning, or it is just the matter of style? How can I repair a buried 120v cable splice to make it meet code? I have an electrical question. I have a cable feeding an outdoor yard light. It was cut, I spliced it, soldered it and wrapped it with tape. Is there a way to splice and bury it safely to code? If not, I'll need to dig a 30 foot trench to replace the whole length.  Was about 8 inches below ground, surfaced over the years and probably hit with the weed trimmer.   It is outdoor cable inside copper tubing. Years ago when I bought the wire I knew outdoor was gray. The box said outdoor but it is white like indoor wire.   It was spliced about 20 years ago but am fixing up house and like to hide the ugly splice. Find $x$ in   $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$   A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} = x^2 \quad\implies\quad x = \pm \sqrt{2} $$  Are these solutions correct? If not, why? If yes, are there other solutions?    PS: An extension of this discussion can be found in Possible bug in sidebar "question asked:" date I want to copy/modify/delete a Minecraft world. Where can I find my saves folder?