How do you integrate the equation $$ \int \frac{(\cos^3x)}{(\sin^2x)}dx $$ using U substitution?
I'm stuck with this integral  $\int\frac {\cos^4x}{\sin^3x} dx$    which I rewrote as  $\int \csc^3x \cos^4xdx$  then after using the half angle formula twice for $\cos^4x$ I got this  $\frac 14\int \csc^3x (1+\cos(2x))(1+\cos(2x))dx$   then after solving those products I got these integrals  $\frac 14 \{\int \csc^3xdx+2\int \csc^3x \cos(2x)dx + \int \csc^3x \cos^2(2x)dx\}$  I do know how to solve the $\int \csc^3xdx$ one but I'm totally lost on the other ones, any tips/help/advice would be highly appreciated! thanks in advance guys!
Seems like a bug:  The user:   Edits a question Then, he decide to close it. Clicks on the Close button. The popup doesn't appear. Refreshes the page. Click on the close button. The popup appears.   After editing in the link appears a hash: #. But I don't think that this is the problem because if I add the hash without editing the close popup appears.  It's minor, but I think that is good to be reported and solved.  Edit: The close popup doesn't open after we click an edit has been made to this post; click to load. My new laptop came preinstalled with Windows 8. Naively, as I am, I just formatted the harddrive and installed fine old Ubuntu. Now I want to install Windows 8 for dual boot again, but I have no DVD and do download the ISO one needs a product key. That key is not on the back of the laptop anymore but somewhere on the mainboard.  Is there any way to recover the product key from the mainboard using Ubuntu? I'm a student of mathematics with not much background in physics. I'm interested in learning Quantum field theory from a mathematical point of view.  Are there any good books or other reference material which can help in learning about quantum field theory? What areas of mathematics should I be familiar with before reading about Quantum field theory? Compactness of $\operatorname{Spec}(A)$ In an exercise in Atiyah-Macdonald it asks to prove that the prime spectrum $\operatorname{Spec}(A)$ of a commutative ring $A$ as a topological space $X$ (with the Zariski Topology) is compact.   Now because the basic open sets $X_f = \{\mathfrak{p} \in \operatorname{Spec} (A) : \{f\} \not\subseteq \mathfrak{p} \}$ form a basis for the Zariski Topology it suffices to consider the case when   $$X = \bigcup_{i \in I} X_{f_i}$$   where $I$ is some index set. Then taking the complement on both sides we get that   $$\emptyset = \bigcap_{i \in I} X_{f_i}^c$$  so there is no prime ideal $\mathfrak{p}$ of $A$ such that all the $f_i$'s are in $\mathfrak{p}$. Now from here I am able to show that the ideal generated by the $f_i$'s is the whole ring as follows. Since there is no prime ideal $\mathfrak{p}$ such that all the $f_i \in \mathfrak{p}$, it is clear that there is no $\mathfrak{p}$ such that $(f_i) \subseteq \mathfrak{p}$ for all $i \in I.$ Taking a sum over all the $i$ then gives $$\sum_{i \in I} (f_i) = (1).$$  Now here's the problem:          How do I show from here that there is an equation of the form $1 = \sum_{i \in J} f_ig_i,$ where $g_i \in A$ and $J$ some finite subset of $I$?       This part has been giving me a headache. I am not sure if the finiteness bit has to do with algebra, topology or the fact that we are dealing with prime ideals.  This is not a homework problem but rather for self-study.  $\textbf{Edit:}$ I have posted my answer below after the discussion with Dylan and Pierre. Lost my old passport for Schengen visa application Once it is compiled, is there a difference between:  delegate { x = 0; }   and  () =&gt; { x = 0 }   ? On some user profiles, e.g.  image proportions are broken:        Image on activity tab or on  page isn’t distorted:        The issue is probably related to the old one: Area of an irregular polygon Writing a compiler in its own language Intuitively, it would seems that a compiler for language Foo cannot itself be written in Foo. More specifically, the first compiler for language Foo cannot be written in Foo, but any subsequent compiler could be written for Foo.  But is this actually true? I have some very vague recollection of reading about a language whose first compiler was written in "itself". Is this possible, and if so how? Proving orthogonality of vectors in a complex vector space I'm taking a course in Real Analysis and have come across the following question which I initially thought looked quite simple:  Let $X$ be an inner product space over $\mathbb{K}$ ($\mathbb{K} = \mathbb{C}$ or $\mathbb{R}$) Show that x and y are orthogonal if and only if $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + ||\mu y||^2$ for all $\mu, \lambda \in \mathbb{K}$.  I have proven the statement in one direction (starting with orthogonality and ending up with $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + ||\mu y||^2$). However, I seem to be getting stuck trying to go in the other direction. The following is what I have so far:  $||\lambda x  + \mu y||^2 = ||\lambda x||^2 + 2\bar{\lambda}\mu Re&lt;x,y&gt; + ||\mu y||^2$  Then we must have 2$\bar{\lambda}\mu Re&lt;x,y&gt; = 0$ for the desired equality to hold, and so $Re&lt;x,y&gt; = 0$.   However, for orthogonality I want $&lt;x,y&gt; = 0$ and can't see how I can make this next step.  Any help would be greatly appreciated. Why are the two versions of Chromium and VLC available in Ubuntu software of different sizes? Feed can show entries for deleted posts, which viewing them fails Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! Video grid with vstack and hstack I'm having trouble restarting a 2D game when the user presses the "New Game" button.  Structure of my game:  The game has several 'screens': Opening screen, instructions screen, main game screen, and a game-over screen.  Each screen is an instance of a class: OpeningScreen class, GameScreen class, etc.  There is a class named Main. This class has three roles.   It's the JFrame (the window) of the program. It has the main method - first class to run. It has a method that change's screens.   When the program starts, this class runs. When it first runs:   Creates a new instance of itself (it's a JFrame-type class). Creates an instance of every 'screen-class'. The constructor of each 'screen-class' takes the instance of Main as an argument: (aka: GameOverScreen gameOverScreen = new GameOver(this)). Creates a new JPanel named mainPanel, and adds it to the JFrame. Any screen to be displayed will be contained in this JPanel (The screens are also JPanel-type classes).   How the changeScreen() method works:  The signature of changeScreen() is this: changeScreen(JPanel from, JPanel to).  Whenever a screen needs to be changed, the current screen that is displayed calls this method, like so: main.changeScreen(this, main.InstructionsScreen). It removes the previous screen from mainPanel and adds the new screen to mainPanel. As said, these screens are created when the program launches.  This is the method that changes screens:   public void changeScreen(JPanel from, JPanel to){      mainPanel.remove(from);     mainPanel.add(to);      mainPanel.revalidate();     mainPanel.repaint();  }   My problem:  After a game is finished, the Game class calls the changeScreen method: main.changeScreen(this, main.GameOverScreen) .  This replaces the game screen with a game over screen. Pressing the "New Game" button in the game-over screen, calls main.changeScreen(this, main.GameScreen) .  The problem: Pressing the "New Game" button in the game-over screen, indeed displays once again the instance of the Game class. But this is the old instance, so the game is not restarted. It's already finished.  I tried to do the following changes in changeScreen():  public void changeScreen(JPanel from, JPanel to){      mainPanel.remove(from);      if(to.equals(gameScreen)){         gameScreen = null;         gameScreen = new GameScreen(this);     }     mainPanel.add(gameScreen);      mainPanel.revalidate();     mainPanel.repaint();  }   This should reset the gameScreen object. But this creates a delay of about 10 seconds when pressing the "New Game" button.  What would be a good way of restarting the game?  Thank you Variance (different definitions) I'm new to statistics and reading stats books I found different definitions for Variance.  Definition1: $ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 $  Definition2: $ s^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 $  I would like to know why is used different definition for same thing? What do Family Tree Maker users do, now that FTM has been discontinued? I was considering purchasing Family Tree Maker only to find it has been discontinued.    What do they plan for a replacement ? Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! how to use dom php parser "viewed 1 times" When a post is first created, it says "viewed 1 times" in the sidebar. This should say "viewed 1 time".