How to prove the following by Cauchy-Schwarz?
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
I have installed some apps through the Ubuntu Software center. How can I find the corresponding terminal command? For many apps, this is not the same as the displayed name. For example, the app called Text Editor can also be launched by typing "gedit" in the terminal, but how can I find this for other apps? Finding cartesian product with PHP associative arrays What level is a spell if you cast it without expending a spell slot? How does the reparameterization trick for VAEs work and why is it important? How does the  for variational autoencoders (VAE) work? Is there an intuitive and easy explanation without simplifying the underlying math? And why do we need the 'trick'? Integrating $\int\frac{x^3}{\sqrt{9-x^2}}dx$ via trig substitution What I have done so far: Substituting $$x=3\sin(t)\Rightarrow dx=3\cos(t)dt$$ converting our integral to  $$I=\int\frac{x^3}{\sqrt{9-x^2}}dx=\int \frac{27\sin^3(t) dt}{3\sqrt{\cos^2(t)}}3\cos(t)dt\\ \Rightarrow \frac{I}{27}=\int \sin^3(t)dt=-\frac{\sin^2(t)\cos(t)}{3}+\frac{2}{3}\int\sin(t)dt\\ \Rightarrow I=18\cos(t)-9\sin^2(t)\cos(t)+c$$ Which is where I'm stuck. I can substitute back in for the $\sin(t)$ terms but don't know hot to deal with the $\cos(t)$ terms and back out information about $x$.   edited: because I miswrote the problem, I am sorry. Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field? During U-substitution, is the $du$ that we use as the infinitesimal in $\int f(u) \ du$ changing size? I am familiar and comfortable with the following definition of an integral (defined as the the Riemann Sum where n approaches infinity).    I have circled in red two terms to emphasize a concept that I was taught: the $\frac{b-a}{n}$ term effectively becomes the $dx$ term. This intuitively makes sense to me.   Additionally, I am familiar with the following statement (as expressed here: )    However, I have boxed in red something that I find rather confusing when trying to interpret it through the understanding that $\frac{b-a}{n}$ sorta kinda equals $dx$...  Specifically, the $du$ value appears to be changing throughout the interval $[ g(a), g(b)]$. To reframe this in the "Riemann Sum" version,we would have something like: $\text{ "value that varies depending on where you are"} * \frac{g(b)-g(a)}{n}$  This is very confusing to me...because, if this is the correct interpretation, I'm not quite certain I understand why this is allowed. Memory allocation for member functions in C++ What part of speech is there in the sentence “There is a book on the table?”  Also, while typing it out, another question pops up vis-à-vis punctuation. In my complete first sentence above, I ended it with a question mark since my main sentence is a question. The quote is not a question, but it looks like it because of the question mark. Is this the right way to punctuate? I have a vps using Ubuntu 12.04 LTS and I my host corrupted my vps so now it won't boot up. When it boots up it says:  Booting from Hard Disk... error: file not found. grub rescue&gt;   How can I fix/repair this? Is there any bidirectional 5v-3.3v level shifter? Is there any IC that interfaces any 3.3V Input/Output to 5V Output/Input? I need it primarily for an Arduino Due but if there any bidirectional IC that works like that it would be great.  Some people advised me to use the SN74AHC125 and CD4050 ICs, but I don't understand how they work or how to interface with them. How to get Beamer Math to look like Article Math What does a two overlapping circles symbol mean in an electrical schematic/diagram? In the datasheet for a A3503 Hall Effect Sensor, I saw a circuit diagram labeled 'Functional Block Diagram', and it shows a symbol I have not seen before. It's shown here between 3 - Output and 2 - Ground.  What is this supposed to represent?    The full datasheet is available Problems with pie charts There seems to be in increasing discussion about pie charts.   The main arguments against it seem to be:   Area is perceived with less power than length. Pie charts have very low data-point-to-pixel ratio   However, I think they can be somehow useful when portraying proportions. I agree to use a table in most cases but when you are writing a business report and you've just included hundreds of tables why not having a pie chart?  I'm curious about what the community thinks about this topic. Further references are welcome.  I include a couple of links:         In order to conclude this question I decided to build an example of pie-chart vs waffle-chart. Is spell level a character concept or a player concept? Is there anything in RAW that supports spell level as a character concept, or alternatively that supports it as solely a player concept?  On the one hand, I see nothing that explicitly states that a character knows that there are spell levels. On the other hand, I don't see how a spellcaster could manage spell slots without knowing about spell levels. How can I get the page title? creating a tar archive without including parent directory I am trying to create a graphical program for my script.  Inside the script I use tar to create a tar archive.  From the graphical program I get the full name of file that I want to create a tar archive.  tar -cvf temp.tar /home/username/dir1/dir2/selecteddir    My tar archive includes home, username, dir1, dir2 and selecteddir while i want tar to create archive only including selecteddir. Can I let my fake flash USB drive get its real capacity through low level format? Recently, I bought an USB flash drive which has a cool design.  The seller said it is 256GB however, I've already known that capacity would be fake. Anyway, I bought it cause of its cool design and want to use this USB flash drive anyway.  What should I do?  I've tried to find a proper tool(vid 1908 / pid 1320) but couldn't. Can I repair this through Low level format? Or partitioning this through Linux? To get the character sequences of a particular keyboard shortcut we use showkey -a or verbatim insert (Ctrl+V). What is the opposite of it? How can I get the keyboard shortcut in human readable format from character sequences? I used some keybindings for example ^[^? ^X^U ^[[3;5~ ^[[1;2D ^[[1;2C ^[[1;5C ^[[1;5D ^[[1;6C ^[[1;6D ....  Now I do not remember which key/key-combination they used to represent. How can I find that out? Update 1 I am looking for something which will output Ctrl+R or Ctrl+Delete etc, I mean in human readable format. Any reference to a table which has possible character sequences and their human readable formats will also do. How Can I View All The Reputation Gained/Lost For One Specific Question If I would like to view all of the reputation I have gained or lost due to one specific question of my own, how can this be done?  Is it possible for someone with, say, 421 rep on the site in question?