I won't be writing each step explicitly because it takes me forever to write this up in math format.  $$ \int_2^4 \frac{\sqrt{\ln{(9-x)}}}{{\sqrt{\ln{(9-x)}}}+{\sqrt{\ln{(x+3)}}}}\,dx $$  My solving of it went as follows: $$ \int_2^4 \frac{\sqrt{\ln{(9-x)}}}{{\sqrt{\ln{(9-x)}}}+{\sqrt{\ln{(x+3)}}}}  \frac{\sqrt{\ln{(9-x)}}}{\sqrt{\ln{(9-x)}}}\,dx= $$ $$ =\int_2^4 \frac{{\ln{(9-x)}}}{{{\ln{(9-x)}}}+{\sqrt{\ln{(9-x)}\ln{(x+3)}}}}\,dx  $$  It has at this point been an hour and I won't waste more time to make the equations look pretty. $$ =\int_2^4\frac{\ln(9-x)}{(\ln(9-x) + \ln(9-x)^{\ln(x+3)/2}}\,dx $$  Factor out ln(9-x) and cancel and you get  $$ \int_2^4\frac{1}{1+\ln(9-x)^{(\ln(x+3)-2)/2}}\,dx $$ After a bit more simplifying (bringing down the exponent and multiplying, then moving the 2 out of the integral, then simplifying the remaining $\ln$'s) you get $$ 2\int_2^4\frac{1}{2+\ln\frac{x+3}{9-x}}\,dx $$ And this is an easy integral because it is directly derived from arctan. I get the antiderivative is $$ \frac{1}{\sqrt2}\arctan\left(\frac{1}{\sqrt2}\sqrt{\ln\frac{x+3}{9-x}}\right). $$ But when I try to use that for the definite integral, I get a negative under the square root, so it must be wrong. Can someone explain where I must've gone wrong? I hope I gave enough information so that you know what I did.
This is a Putnam Problem that I have been trying to solve (on and off) for two years, but I have failed. I am in Calculus BC. This problem comes from the book "Calculus Eighth Edition by Larson, Hostetler, and Edwards". This problem is at the end of the first section of the chapter 8 exercises. Here's the problem:  Evaluate $$\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)} + \sqrt{\ln(x+3)}}.$$  Please. Any help is very much appreciated. So are solutions. Thank you!  Edit: I like the solution given, but I was interested to see if there is any other way of doing the problem? I'm excited to see the results.
Underscores in words (text) How can I produce the text Word_one_two in LaTeX?  I tried:  Samp\_Dist\_Corr   But, it doesn't quite look right. Also, I want it in the typewriter font, so actually, I'm doing:  \texttt{Samp\_Dist\_Corr}   I find it looks a bit like the underscore is merging in to the bottom of the "D", but maybe it's just because of the typewriter D? What is the difference between Affection and Friendship, if any? I've risen Pokemon Amie Affection to 5 hearts on my Riolo in the hopes that it will evolve as soon as it levels up, but I'm not certain that this is the same thing as friendship, and I'd like to not waste a level.  My question is this: What is the difference between Pokemon Amie Affection and the Friendship stat? How to make unit selection circles merge? I would like to know how to make this effect of merged circle selection. Here are images to illustrate:     Basically I'm looking for this effect:    How the merge effect of the circles can be achieved ? I didn't found any explanation concerning this effect. I know that to project those texture I can develop a decal system but I don't know how to create the merging effect.  If possible, I'm looking for purely shaders solution. What is the difference between implication symbols: $\rightarrow$ and $\Rightarrow$? For example how come $\zeta(2)=\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}$. It seems counter intuitive that you can add numbers in $\mathbb{Q}$ and get an irrational number. Manually installing a program on Debian ${{2\pi i=0}}$? I was trying to solve for $i^i$.  I got distracted and did the following: $e^{i \theta} = \cos(\theta) + i\sin(\theta)$ $e^{i 2\pi} = cos(2\pi) + i\sin(2\pi) = 1$ ||| Take ln() of two sides of equation: $\ln(e^{i2\pi}) = \ln(1)$ $i2\pi = 0$ Clearly this is false. What did I do wrong? Workflow for UV orientation How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 What are the consequences of "copying" a data set for OLS? Algebraic dual of $\mathbb{R}^{\mathbb{N}}$ As in the title, my question is: What is the algebraic dual space of $\mathbb{R}^{\mathbb{N}}$ - the space of all $\mathbb{R}$-valued sequences. I think that it should be the direct sum $\bigoplus_{\mathbb{N}} \mathbb{R}$, via the natural duality, but I cannot proof it. How to run/use popcorn-time? I got popcorn time installed using this link:    But I have no clue how to use or run Popcorn time.  Do I open my torrent client and download a movie and then it gets going? or do I have to open it? If I have to open it, how do I do that, I opened the dash and tried clicking on it but nothing happened. Please help me!! :D Should form 'Continue' button be disabled if validation is incomplete? In forms we often see the 'continue' button inactive until all the required fields are complete:         &ndash; Wireframes created with   Is this actually a help to the user, or a hindrance?  I imagine the Pro for this is that it's a visual indicator to the user that they haven't finished filling in the form or dealing with errors (because even with inline validation a user won't be informed if they've never actually interacted with a field so won't show any errors for those at that point).  But then the main Con for this (which I've also seen borne out in usability testing) is that users will be confused why they can't click the button.  Is the pattern of disabling the Continue button until validation is complete actually providing a negative user experience? What new capabilities do user-defined literals add to C++? introduces  which will allow the introduction of new literal syntax based on existing literals (int, hex, string, float) so that any type will be able to have a literal presentation.  Examples:  // imaginary numbers std::complex&lt;long double&gt; operator "" _i(long double d) // cooked form {      return std::complex&lt;long double&gt;(0, d);  } auto val = 3.14_i; // val = complex&lt;long double&gt;(0, 3.14)  // binary values int operator "" _B(const char*); // raw form int answer = 101010_B; // answer = 42  // std::string std::string operator "" _s(const char* str, size_t /*length*/)  {      return std::string(str);  }  auto hi = "hello"_s + " world"; // + works, "hello"_s is a string not a pointer  // units assert(1_kg == 2.2_lb); // give or take 0.00462262 pounds   At first glance this looks very cool but I'm wondering how applicable it really is, when I tried to think of having the suffixes _AD and _BC create dates I found that it's problematic due to operator order. 1974/01/06_AD would first evaluate 1974/01 (as plain ints) and only later the 06_AD (to say nothing of August and September having to be written without the 0 for octal reasons). This can be worked around by having the syntax be 1974-1/6_AD so that the operator evaluation order works but it's clunky.   So what my question boils down to is this, do you feel this feature will justify itself? What other literals would you like to define that will make your C++ code more readable?    Updated syntax to fit the final draft on June 2011 What are suitable TV sizes and viewing distances for couch surfing on a 1080p LCD HDTV? How to change the style of the title attribute inside an anchor tag? Why can I not access a file named `-` when quoting? I have a file called -.  I want to display its contents.  One way is to do  cat ./-   since cat - reads from standard input. However, why are cat "-" and cat '-' also interpreted by the shell as cat -? File sync error (AUTH_FAILED) Recently I started getting a File sync error on all 3 of my PCs running Ubuntu One.  One is running Ubuntu, the other 2 are running Windows 8.  Looking at the list of clients on the web, none of them were there anymore.    I found this article:      On how to fix the issue for my Ubuntu PC and it worked perfectly.  Now I am looking for a similar fix for Windows but haven't found one yet.  I've tried un-installing Ubuntu One and re-installing it, but it must still be holding the keys somewhere because it never asked me to login again.    Does anyone know where I can clear these keys to get my files syncing on Windows again? In comments, moderators have their diamonds in the OP rectangle:        And in the review (suggested edits), mods have their diamonds outside the blue OP background highlight rectangle:        Of course, it Doesn't Really Matter At All, but it is a small discrepancy (I am not saying that in the review, OP rectangles have a bigger padding). Prove that $T$ has an orbit of period 3 of Stack Overflow, all content posted on Stack Exchange sites by their users (i.e. you wonderful people) has been provided to the whole universe under . For my fellow non-lawyers, that license basically means:   Anyone can use any Stack Exchange posts at any time without having to ask for permission Making money off of the copied content is permitted You don't even have to copy stuff from here verbatim; you can just use it as a starting point and make whatever edits you want There are just two rules you have to follow:   You have to provide . Simple links to the original post and author info are just fine. You have to  and allow other people to use your content, as long as they follow these very same rules. How meta!    (If you ever forget any of that, and want to refresh your memory, the license info is linked to in the footer of every page.)  There are, in fact, a lot of people who republish varying amounts of our content for assorted reasons. Unfortunately, there are some Stack Content Republishers Attributing Poorly and/or Excelling at Ranking (SCRAPERs, for short).  In this context, "attributing poorly" means any use that doesn't follow our attribution rules or make any other reasonable attempt at give credit. This can get pretty egregious; I've seen SCRAPERs that not only don't link back to SE originals, but also use fake author info and post dates to make it harder to find originals. By "excelling at ranking," I'm referring to copycat sites that end up higher in Google results than the original SE sites do for the same content. That's not necessarily wrong, but in some cases, it indicates inappropriate SEO hackery.  So, the question is: what can you do if you spot a SCRAPER?