Suppose $f'$ is continuous and positive on $[a, b]$. Prove $\int_a^b f(x) dx + \int_{f(a)}^{f(b)} f^{-1} (y) dy = b f(b) - a f(a)$
Prove that:$\int_{a}^{b} f(x) dx = b \cdot f(b) - a \cdot f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) dx$
Lie Algebra of $SL_n(\mathbb H)$ The Lie algebra of $SL_n(\mathbb C)$ are the matrices where the trace is $0$. But what is the Lie algebra of $SL_n(\mathbb H)$ where $\mathbb H$ is the quaternions? When selecting components, is there a way to control the "selection distance/area"? I don't want to select a component, even if I click ~20 pixels away from it.    Thanks in advance! Show that the boundary of a set equals the boundary of its complement $\newcommand{\bdy}{\operatorname{bdy}}$  I'm trying to show that $\bdy(A) = \bdy(A^c)$.  I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know where to go from there.  Any help or hints would be very much appreciated. When is JavaScript's eval() not evil? I'm writing some JavaScript code to parse user-entered functions (for spreadsheet-like functionality). Having parsed the formula I could convert it into JavaScript and run eval() on it to yield the result.  However, I've always shied away from using eval() if I can avoid it because it's evil (and, rightly or wrongly, I've always thought it is even more evil in JavaScript, because the code to be evaluated might be changed by the user).  So, when it is OK to use it? What is the purpose of delayed suspend (Ctrl-Y) in Bash? I figured out how to use a normal image as the background while using an HDRI for environmental lighting. But I can't figure out how to use a different HDRI as the background.  Please note I already read this answer:  but this appears to be different for Eevee in 2.8.  Thanks! How to install Anaconda on Ubuntu? How to install  on Ubuntu?   Is there a way to use apt-get install?   I only have command line access to my server. How do I install Anaconda on Ubuntu 14.04 from the command line? In an enumerate environment, I want a few (not all) of the items to have asterisks which appear before the number (I don't feel strongly about whether the number or the asterisk is horizontally aligned with the other numbers).  Is there a less crufty way to do this than, say, using multiple lists (or &mdash; horrors &mdash; numbering manually)?  I'm already using enumitem for an unrelated purpose, if that helps, although I didn't find a solution after a quick skim of its manual. I asked about it on  and they told me to post it here.  Probably need to add the :active to the :hover meta class or something like that.  Example:      Hammer Time File names for JSP include directive to avoid compilation of them Word or expression for a constituent that perfectly exemplifies the whole Suppose that  (1990) is considered appalling, but you can't put your finger on exactly why, mainly because there are a plethora of reasons, most of which you lack the training/education/expertise to comment in-depth on (e.g. the writing, directing, editing etc.), so you use what you consider to be a disastrous moment/scene as an example, and think "this pretty much sums it up", for example , what would you call a scene like that? A single word or expression is satisfactory.  As an example:     "The Godfather 3 is so bad, for so many reasons, but here is a ______ that perfectly represents everything wrong with the movie."   An expression is also acceptable, and the example I gave is flexible (doesn't have to be structured as it is in the example). Can you add new statements to Python's syntax? Can you add new statements (like print, raise, with) to Python's syntax?  Say, to allow..  mystatement "Something"   Or,  new_if True:     print "example"   Not so much if you should, but rather if it's possible (short of modifying the python interpreters code) Ready to use leaderboard service. Is any? I'm looking for ready to use thirdparty leaderboard service where player's achivementes and scores can be placed. I'd intagrate it into my game to allow players push scores and locate themselves in the tables of tops. It would be nice whether anybody experienced share some points with me. Force newlines with cat wildcard printing I want to use cat with wildcards in bash in order to print multiple small files (every file is one sentence) to standard output. However, the separate file contents are not separated by a newline, which I'd like to for ease of reading.  How can I add a file delimiter of some sort to this command? "Improve" treated as improper response to proper response to audit Can I store all the content of my previous iPhone in a new iPhone using iTunes? How to get Url Hash (#) from server side I know on client side (javascript) you can use windows.location.hash but could not find anyway to access from the server side. I'm using asp.net. $_SERVER['QUERY_STRING']; not returning full query string Where does the extra kinetic energy come from in a gravitational slingshot? I read in  in this site that the KE a free-falling ball acquires is not originated by the attracting body but that energy was actually stored in the ball when it had been lifted to the height it dropped from.  In this way, it was said, gravity is subject to the conservation of energy principle and cannot change the total energy of an object.  Consider now the maneuver known as a  (also gravity assist) used by space probes such as the . A space probe approaches a planet with velocity $v$, slingshots around, and ends up with velocity $v+2U$, where $U$ is the velocity of the planet.  Consider the energy of the probe. Before, it was $E_i=\frac{1}{2} mv^2$ and after it is $E_f=\frac{1}{2} m(v+2U)^2$. It looks like $E_f$ is much bigger than $E_i$ - but where did the additional energy come from?   Is this not a violation of the conservation of energy principle? What are SPF records, and how do I configure them?