$$\int \sin^2 x \cos^3 x dx$$  $$\int (1-\cos^2 x) \cos^3 x dx$$  I know there is a rule in my book (with little explanation) that tells me when I had an odd and an even degree on two trig functions I should split the odd and convert it to an identity but this way seems easier, and I can't get an answer either way.  $$\int \cos^3 dx - \int \cos^ 5 x dx$$  I am not sure where to go from here, I don't know how to get the integral of $\cos^3 x$
Is this correct? I thought it would be but when I entered it into wolfram alpha, I got a different answer.   $$\int (\cos^3x)(\sin^2x)dx  = \int(\cos x)(\cos^2x)(\sin^2x)dx  = \int (\cos x)(1-\sin^2x)(\sin^2x)dx.$$  let $u = \sin x$, $du = \cos xdx$  $$\int(1-u^2)u^2du = \int(u^2-u^4)du  = \frac{u^3}{3} - \frac{u^5}{5} +C$$  Plugging in back $u$, we get $\displaystyle\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5}$ + C
How well does T-Mobile's (US) Simple Choice plan work when in Europe? Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? Copy files from Network Mapped Drive I am trying to copy a file over from a mapped drive on a network to the current computer through a batch file.   I've already attempted the following below and still no results. The plan is to run a batch file to copy a file from a network drive down to the local C:\ drive.   @echo off copy H:\Videos\TargetVideo\TargetVideo.mp4 C:\Video\ Is there a way to transfer money and items from iOS to Android? I've been playing Jetpack Joyride on iOS, and I just got an Android device. Can I move my things, such as money and unlocks, from my iOS device to Android? Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices? I read in the book Applied Analysis by Hunter and Nachtergale that     the sequence $x(n)=\log(n)$ is not Cauchy since $\log(n)\to\infty$   But that seems to be irrelevant to the definition of a Cauchy sequence which I understand is as follows:     A sequence $x(n)$ is said to be Cauchy if for every $\epsilon &gt; 0$  there exists an $N$ such that $\lvert x(m)-x(n)\rvert &lt; \epsilon$ for all $m,n&gt;N$.   This sequence $(\log n)$ seems to meet the definition. So how come it is not considered Cauchy? How do you name your ViewModel classes? How did Asuna and Kirito survive? In the last episode of Aincrad arc of Sword Art Online, how did Asuna survive even though her avatar disintegrated after being attacked by Kayaba? Also Kirito was stabbed by Kayaba as well. How did he survive?  Was it in the novel on how they really survived? How do I get a platform-dependent new line character? How do I get a platform-dependent newline in Java? I can’t use "\n" everywhere. Greek pdf bookmarks I am puzzled with sections titled with Greek characters. They do appear in the pdf's Content list, though not correctly.  For example, in my thesis--written in English with LyX, based on the KOMA-script--I choose to include a translation of the summary in Greek (=Περίληψη). The "raw" code for the (unnumbered) chapter title reads: \addchap{\textgreek{Per\char208lhyh}}. In the final .pdf file (using pdflatex) the respective entry appears as Per208lhyh. Using the unicode option for the hyperref package does not fix the problem.  Update detail: the document (should) not only contain Chapter or Section titles in Greek, but normal text as well.  I have found some relevant references within tex.stackexchange and elsewhere (e.g., , the manual of the textgreek package, , ). However, I cannot extract a concrete answer.  How can Greek characters appear correctly as bookmarks in the .pdf's Content list?  Note, the information of this Q&amp;A applies to/under LyX as well. Appending to an ObjectOutputStream What if I offer a bounty and don't get any good answers? I have started a bounty:   It expires in 3 days. I don't have an appropriate answer to accept, and the only answer which it has gotten has no votes.  So am I going to simply lose 100 reputation without getting a satisfactory answer?  Or should I accept the lone inappropriate answer so that I will not lose my right to accept answers to any questions which I might ask in the future, which is what the bounty rules claim? Finitely generated group which is not finitely presented Is there any easy group theoretical way of showing that the wreath product $G$ of two infinite cyclic groups is not finitely presented?  I was looking for a finitely presented group with a central subgroup isomorphic to the free group of countable rank and whose factor group is isomorphic to $G$; in this way assuming that $G$ is finitely presented we get a contradiction.   Any ideas? Since one or two months ago, my laptop with ubuntu 12.10x64 hangs during shutdown. There's just a black screen, no harddisk activity, it justs sits there. I feel it happens near the end of a normal shutdown cycle, judging from time elapsed.  Only resolution is hard power-off (holding power key for several seconds) - neither Ctrl-Alt-Backspace (which normally works, I have it switched on in Settings) nor Ctrl-Alt-Del help, nor can I get into a text console with Ctrl-Alt-F1.   How do I diagnose that? It does not happen on every shutdown, but often enough to be annoying (as it forces a diskcheck on reboot, of course).   Which log files can tell me what's going wrong? More or less blindly searching in the sys log viewer reminds me of needles in a haystack.  I imagine I could pinpoint the time it happens via log timestamps, but how do I read timestamps like e.g. in kern.log: [36169.146130]? How do I get this human-readable?  Lastly, a bit of a btw, I've never managed to execute an Alt-Sysrq-REISUB to force a cleaner shutdown, but that's probably because I'm on a laptop, PrintScr is only accessible via an Fn key, and I'm not sure of the exact key sequence I need. Is there a "harmless" but user-visible magic command I could practice this with (during normal operation), so I can find out how to do a REISUB next time?  Thanks all! Comment if you need additional data... How to make a persistent live Ubuntu USB with more than 4GB I want to make a persistent live Ubuntu USB drive with more than 4GB of storage.   I know there is an option of installing directly on the drive but I don't want to do this as it will restrict the drive from working on different computers.   One option is to make an additional partition with extra space but if I do this how can I get newly installed software to be stored on this partition? How to make mesh lights work in Blender Eevee? The closest I can get to making mesh lights affect their environment in Blender Eevee is giving them an Emission material and turning on "Bloom". But this isn't emitting proper light, it just looks like it is, but it doesn't light the surroundings whatsoever.     I would like to be able to light a scene with meshes without having to put point lamps everywhere just to make them emit light. Cannot translate vertice/edge will duplicate Most of the time when I try to move a vertex or an edge, instead of translating that, it extrudes it, it creates another face or, vertiex or whatever. I do not really get why it behaves like that. I hope I explained it in depth. I am sorry if this is a duplicate post but I could not find anything! How do I remove/delete a folder that is not empty? I am getting an 'access is denied' error when I attempt to delete a folder that is not empty. I used the following command in my attempt: os.remove("/folder_name").   What is the most effective way of removing/deleting a folder/directory that is not empty? I understand test -n &lt;expression&gt; to return false if the expression evaluates to a string length of greater than 0.  Why then does the following happen?  Macbook:~ echo ${#undeclared_variable} 0 Macbook:~ test -n $undeclared_variable &amp;&amp; echo Foo Foo   Bash 4.x  My expectation is that since test -n $undeclared_variable should evaluate false, that Foo should not be echoed.  If I use [[  ]] instead of test, it works as expected. Suppose that sequences of real numbers satisfy:  \begin{align*}   a_n &amp;= \sum\limits_{r=0}^n \binom{n}{r} b_r \\ \end{align*}  Prove that:  \begin{align*}   (-1)^n b_n &amp;= \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s \\ \end{align*}  My work:  The $n=0$ case:  \begin{align*}   a_0 &amp;= b_0 \\   b_0 &amp;= a_0 \\ \end{align*}  Then $n=1$ case:  \begin{align*}   a_1 &amp;= \binom{1}{0} b_0 + \binom{1}{1} b_1 = b_0 + b_1 \\   -b_1 &amp;= \binom{1}{0} a_0 - \binom{1}{1} a_1 = a_0 - a_1 \\   b_1 &amp;= a_1 - a_0 = b_0 + b_1 - a_0 = b_1 \\ \end{align*}  The inductive step:  \begin{align*}   (-1)^{n+1} b_{n+1} &amp;= \sum\limits_{s=0}^{n+1} \binom{n+1}{s} (-1)^s a_s \\ \end{align*}  I'm not sure where to go from here. I tried substituting $a_s$ into that last equation and it didn't help. Any ideas?