I am curious about the proof of these two integrations. It would be great if you could point me in the right direction : $ \int{e^{ax}\cos({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\cos(bx)+b\sin(bx)] + C $ &amp; $ \int{e^{ax}\sin({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\sin(bx)-b\cos(bx)] + C $
Is there a general formula for finding the primitive of $$e^{ax}\sin(bx)?$$  I've done this manually with $a=9$ and $b=4$ using Euler's formulas. But it takes a bit of time. Is there a pattern here?
Estimating partial sums $\sum_{n = 1}^m \frac{1}{\sqrt{n}}$ Apostol's Calculus, exercise number I 4.7 13.     Prove that if $n \geq 1$, then   $$ 2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1}) $$   and use this to prove that if $m \geq 2$, then   $$ 2 \sqrt{m} - 2 &lt; \sum_{n = 1}^m \frac{1}{\sqrt{n}} &lt; 2\sqrt{m} - 1 $$   In particular, when $m = 10^6$ the sum lies between $1998$ and $1999$.   After establishing the trivial case, for $n=1$, I can't think of a way to perform the inductive step:  assuming $2(\sqrt{n+1} - \sqrt{n}) &lt; \frac{1}{\sqrt{n}} &lt; 2(\sqrt{n} - \sqrt{n-1})$ and using this, deducing the same for $n+1$ is true as well:  $2(\sqrt{n+2} - \sqrt{n+1}) &lt; \frac{1}{\sqrt{n+1}} &lt; 2(\sqrt{n+1} - \sqrt{n})$  Any help is very appreciated, thanks I have a directory that I am trying to clean out that contains both files and subdirectories.  What I am trying to do is simple: move all the files into another directory, but leave all the sub-directories as they are.  I am thinking something like:  mv [*_but_no_dirs] ./other_directory   It seems like there should be a simple way to do this with wildcards * and regex...  Anyone have ideas? Some applications simulate a virtual USB or CD Rom drive as if a USB drive is attached to the computer.  Is there any configuration or application that provides a virtual USB drive, not for the the operating system itself, but for other equipments which accept USB drive, through a USB port.  So I'll have a virtual hard disk (e.g. a *.vdi file) in the computer, which is connected, through a USB socket, as a USB drive to some other equipment (e.g. a cell phone or a laptop). Choosing numbers without consecutive numbers. In how many ways can we choose $r$ numbers from $\{1,2,3,...,n\}$,   In a way where we have no consecutive numbers in the set? (like $1,2$) java.lang.IllegalArgumentException: Invalid  in servlet mapping &lt;servlet&gt;     &lt;servlet-name&gt;myservlet&lt;/servlet-name&gt;     &lt;servlet-class&gt;workflow.WDispatcher&lt;/servlet-class&gt;     &lt;load-on-startup&gt;2&lt;/load-on-startup&gt; &lt;/servlet&gt;  &lt;servlet-mapping&gt;     &lt;servlet-name&gt;myservlet&lt;/servlet-name&gt;     &lt;url-pattern&gt;*NEXTEVENT*&lt;/url-pattern&gt; &lt;/servlet-mapping&gt;   Above is the snippet from Tomcat's web.xml. The URL pattern *NEXTEVENT* on start up throws     java.lang.IllegalArgumentException: Invalid &lt;url-pattern> in servlet mapping   It will be greatly appreciated if someone can hint at the error. ­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­­ How do I change the default file manager back to Nautilus? A while back, I set my default file manager to Nemo. I like it a lot, but it constantly crashes. I remember editing a text file to set it as default, but I forgot which file it was. How do I switch back to the Nautilus file manager? How to set up beamer to grey out past items but hide future items? pygame.event.get() not returning any events when inside a thread How to find the most common name in passwd file Is it okay to copy-paste answers from other questions? This one leaves me with a bad taste in my mouth, but I defer to others as to whether it is appropriate or not. In , I wrote an answer to a question based on my experiences; it was upvoted and accepted, woo.  Now, today, in response to a different &#8212; but related &#8212; , someone has pasted my answer nearly verbatim (awkward English intact, I may add) as an answer.  Is that really okay? Is it a behavior that should be encouraged? Discouraged? how to host multiple pages from diferent domains all together? How to apply a modifier in V2.9+ Why I can't apply mirror modifier? Here's screen Is it a good practice to show a sign-up or registration form as seperate parts rather than in one go? I am designing a registration form for a site with multiple sets of information required from the user. I have categorized the data fields into relevant groups. Now should I present the form sections in one go or follow a wizard-like approach? What things should I keep in mind while deciding on this? Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.   I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$  I am trying to go ahead of this step. Implications of parity violation for molecular biology In biology, the concept of parity emerges in the context of chiral molecules, where two molecules exist with the same structure but opposite parity. Interestingly, one enantiomer often strongly predominates over the other in natural biological systems (e.g. D-glucose is ubiquitous, L-glucose is rare in nature).  In physics, it has been established that weak interactions violate parity, which (if I understand correctly) implies a physical difference between left- and right-handed systems.  This suggests to me the following questions: Does parity violation in physics affect the physical or chemical properties of chiral molecules? And does this have any implications for our understanding of biological homochirality? MiKTeX: Two things did not succeed I installed a new hard drive on my computer this summer and I'm trying to install MiKTeX on it now. There were no problems when I was using my old hard drive. My OS is Windows 8.1. I'm trying to compile the following very basic file with pdfLaTeX:  \documentclass{article} \begin{document}   hi   \end{document}   Here's the first error I get:  Sorry, but "MiKTeX Configuration Utility" did not succeed.     The log file hopefully contains the information to get MiKTeX going again:   C:/Users/Tyler/AppData/Local/MiKTeX/2.9/miktex/log/initexmf.log  You may want to visit the MiKTeX project page (http://miktex.org), if you need help.   With the following errors in the log:  2015-09-03 22:23:51,625-0400 INFO  initexmf - starting: MiKTeX Configuration Utility 2.9.5700 (MiKTeX 2.9 64-bit)    2015-09-03 22:23:51,631-0400 FATAL initexmf - Windows API error 32: The process cannot access the file because it is being used by another process. 2015-09-03 22:23:51,631-0400 FATAL initexmf - Info: C:\Users\Tyler\AppData\Local\MiKTeX\2.9\miktex/data/le\1e0c0e04cb3be262fbb8e6e7ace14b67.fndb 2015-09-03 22:23:51,631-0400 FATAL initexmf - Source: Libraries\MiKTeX\Core\win\winFile.cpp 2015-09-03 22:23:51,631-0400 FATAL initexmf - Line: 520   The second error:  Sorry, but C:\Program Files\MiKTeX 2.9\miktex\bin\x64\miktex-pdftex.exe did not succeed. The log file hopefully contains the information to get MiKTeX going again:   C:/Users/Tyler/AppData/Local/MiKTeX/2.9/miktex/log/miktex-pdftex.log  You may want to visit the MiKTeX project page (http://miktex.org), if you need help.   with the following errors in the log:  2015-09-03 22:16:18,715-0400 INFO  miktex-pdftex - running 'initexmf --quiet --update-fndb' to refresh the file name database 2015-09-03 22:16:18,746-0400 FATAL miktex-pdftex - The operation failed for some reason. 2015-09-03 22:16:18,746-0400 FATAL miktex-pdftex - Info: C:\Program Files\MiKTeX 2.9\miktex\bin\x64\initexmf.exe 2015-09-03 22:16:18,746-0400 FATAL miktex-pdftex - Source: Libraries\MiKTeX\Core\Process.cpp 2015-09-03 22:16:18,746-0400 FATAL miktex-pdftex - Line: 290   Things I've tried:   Uninstalling and reinstalling MiKTeX Downloading all packages whose name starts with miktex (which is a solution that worked for someone else in a different thread) Having nothing open but TeXworks when trying to compile Looking for processes that may be related to MiKTeX in the task manager (I didn't find any)   How can I stop these errors? I apologize if my formatting is bad; this is my first stackexchange question. Thank you for your help. There does not exist any integer $m$ such that $3n^2+3n+7=m^3$ I have this really hard problem that I am working on and I just don't seem to get it. The question is: let $n$ be a positive integer; prove that there does not exist any integer $m$ such that $3n^2+3n+7=m^3$.  Please explain in detail otherwise I might not understand. I am also open for any methods that might help. Thank you. Please help. Cardinality of the set of bijective functions on $\mathbb{N}$? An idiom meaning someone's doing something useless and has no result at the end In my native language, we use an idiom to warn someone that they're doing something which has no result at the end:     Trying to convince him is like squashing water ...   Is there any idiom in English with the same meaning? How do I have a camera follow my object in Unity? I have an object that automatically moves by itself and I want the main camera to automatically follow it.(Like in games such as geometry dash and jetpack joyride)  This is the code for the automatic moving object in case it is needed:  using UnityEngine; using System.Collections; public class automove : MonoBehaviour { public static int movespeed = 5; public Vector3 userDirection = Vector3.right;    public void Update() {     transform.Translate(userDirection * movespeed * Time.deltaTime);  } }   So does anyone know any good scripts I could add to the main camera to follow this object that automatically moves? Thanks in advance!