I was trying to prove the identity:  $$\int_{0}^{\pi} {xf(\sin(x))} dx =  \frac{\pi}{2} \int_{0}^{\pi} {f(\sin(x))}$$ Firstly, I substitute $x=\pi - t$. Then: $dx=-dt$ and substitute that into the integral, I get: $$I=-\int_{\pi}^{0} \pi f(\sin(\pi - t)) - tf(\sin(\pi-t))dt.$$  After that I get: $$I +\int_{0}^{\pi} tf(\sin(t)) dt=\int_{0}^{\pi} \pi f(\sin(t)) dt $$ So, can I factorize the LHS of the equation to let it become 2I, and I can prove the expression, if it can be factorized, why? The variables are different.   Thank you very much for your reply
Let $f(\sin x)$ be a given function of $\sin x$.   How would I show that $\int_0^\pi xf(\sin x)dx=\frac{1}{2}\pi\int_0^\pi f(\sin x)dx$?
What are the rules about take a bike on the London Tube? Can I take a bike on the Tube (underground) in London, if so how much does it cost? Gauss's Law with Moving Charges Is $0$ a natural number? Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers. If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ How do I prepare best for my first wedding photography event? I am going to shoot a wedding this weekend, and I want to do my best for my friend's big day.    What can I do to prepare for the day in order to minimize failures and mistakes? How do I avoid missing important shots? What backup plan should I have?   I have been shooting for 6 years or so, and I know wedding is one of the hardest thing to do. I have told my friend what I end up producing will be of no match with the work of dedicated wedding photographers, not even the ones that are about to fall out of business.  They are on a tight budget, they said some photos are still better than none, they really must minimize the cost as much as they can. I exaggerated again that my photos will be worse off than the cheapest wedding photographer you can find.  She has been a close friends for years, I want to do everything I can to help her.  Their wedding is at 12:30pm coming Sunday, I have visited the venue last Sunday at 12:30pm to check traffic condition and also the lighting.  The setup is mainly outdoor (no church), the indoor area will not have a good ceiling for bounce. Being at the middle of the day, this can mean some tricky fill flash. I have jot down a list of must-have shots. I am also asking for the seating plan and the event run-down.  Gear:   Canon EOS 60D , EF-S 15-85mm f/3.5-5.6 IS USM Canon EOS 450D (or I can borrow a 550D), EF 50mm f/1.4 Speedlite 580 EXII Speedlite 270 EXII Yongnuo 460 (a cheap but powerful manual flash as backup) Samsung EX-1 f/1.8 point and shoot (backup of a backup...?)   Other lenses that I have but I don't think will be useful:   50mm f/1.8 EF-S 50-250mm f/4.0-5.6 IS II EF-S 18-135mm f/3.5-5.6 IS (I have the better 15-85 on 60D, should I put this on the 450D as a backup allrounder?) EF-S 60mm f/2.8 Macro (should I replace 50mm f/1.4 with this for added macro? You can hardly use f/1.4 under direct sunlight if you want to have fill-flash)   I have two batteries for all cameras (60D 450D and EX1). I have like in total 64GB SD card (I am really inclined to get more, or bring a laptop for on-the-fly backup, or do both). I recently repaired my 580EXII, it has a brand new flash head so the recycle time is as good as it can be, I just need a tonne of batteries. No I don't have external power pack or battery grip for my cameras.  I have a great monopod, but a cheap tripod.  I am happy to bring everything, just putting them in the boot of my car. The carpark is on-site, walking from venue to my car is under 1 minute.  I will have my girlfriend carry the 450D and shoot details / non-portraits.  One concern that I have is fill flash, I really do not have a large surface to bounce off  from. I have shot events before but they are not as dynamic/critical as a wedding.  How do I prepare for differing weather conditions?  If its sunny, fill flash is probably going to be in every single shot. If its cloudy, I probably will turn off the flash. If it rains, my 60D should be fine I have used it in slight rain before, but I really don't know their plan and if they have extendable shelters etc.  What else should I prepare for? Having no experience I want to be as prepared as I can be. Maybe I have overlooked something, maybe there is something else I can do to ensure the shooting will go smoothly. Your help is greatly appreciated! Frequently loaded packages: Differences between pdfLaTeX and XeLaTeX Until now, I have been compiling my LaTeX documents with pdfLaTeX. My standard document preamble includes the "usual suspects" for non-English speakers (, , ) and a bunch of other packages. Assuming I were to convert to XeLaTeX, what modifications of my preamble are advisable? I'm aware that the expansion=true option of the  package is not (yet) available for XeLaTeX, and that  is sort of a default package for OpenType fonts. What other packages (and package options) should be removed and, vice versa, incorporated when switching from pdfLaTeX to XeLaTeX? When you call the object.__repr__() method in Python you get something like this back:    &lt;__main__.Test object at 0x2aba1c0cf890&gt;     Is there any way to get a hold of the memory address if you overload __repr__(), other then calling super(Class, obj).__repr__() and regexing it out? How to write (La)TeX (with parentheses) [or any other TeX-related logo] Where is the right place to register/enqueue scripts & styles Lines too long due to text inside custom command not being hyphenated I'm having some problems with lines that turn out to be too long. There are correct hyphenation rules for the word but they aren't being applied.  Let me first give the sentence that is too long:  An isomorphism that maps a graph to itself is called an \concept{automorphism}.   The command concept is defined as follows  \newcommand{\concept}[1]{\index{#1}\marginpar{\raggedright\textbf{#1}}\textbf{#1}}   The problem seems to be that words inside \concept aren't being hyphenated. How can I fix this? If I type   An isomorphism that maps a graph to itself is called an \textbf{automorphism}.   then the word is hyphenated perfectly. So I would now like to have the command \concept behave the same way, i.e. allow concepts to be hyphenated according to the standard rules. Read echo'ed output from another PHP file Is my expansion of the state $| x \rangle$ correct? Sum of reciprocals of numbers with certain terms omitted I know that the harmonic series $1 + \frac12 + \frac13 + \frac14 + \cdots$ diverges. I also know that the sum of the inverse of prime numbers $\frac12 + \frac13 + \frac15 + \frac17 + \frac1{11} + \cdots$ diverges too, even if really slowly since it's $O(\log \log n)$.  But I think I read that if we consider the numbers whose decimal representation does not have a certain digit (say, 7) and sum the inverse of these numbers, the sum is finite (usually between 19 and 20, it depends from the missing digit). Does anybody know the result, and some way to prove that the sum is finite? I have two independent groups: from one I have drawn a sample of size less than 30 due to its small population, while the second group has sample size 100.  Is it appropriate to use an independent samples t-test in this case? I am designing the layout for a touchscreen for outside use, What are some good color combination to use that will be visible under the sun light for outside use? Maybe bright neon colors? Trichromelibrary was suddenly downloading from Google Play Store From nowhere, my friend got a notification that com.google.android.trichromelibrary was downloading together with a progress bar.  Why? Just need help understanding this question, as I don't really understand what a continuous random vector is. My limited understanding is that a continuous random vector must be completely continuous (so for continuous X and Y this is satisfied) and that to get the probability of the random vector occurring, we double integrate over the supports (of X and Y obviously). My guess is then, that the c.r.v (X,Y) IS a continuous random vector.  Also, what of the converse? If (X,Y) is a continuous random vector, then are X and Y continuous random variables? I also think this is correct, as the continuity of the vector MUST imply the continuity of X and Y, although I'm honestly not sure.  Thanks What is the difference between encrypting something and hashing something?  in what situations would I want one or the other? Windows 7 msconfig lets users select the options No GUI boot and OS boot information to have a list of drivers being loaded during boot. Is it possible to have similar detailed messages during shutting down/restarting/logging off? Here's the screen I want disable or give me more details: What will be total number of solutions of $a+b+c = n$? Please tell me how to find the total number of intergral solutions of $$ a+b+c=n $$ I already know that total number of solutions will be $(n+3-1)c(3-1)$.  But what will be the case when a varies from let $2$ to $6$ , $b$ varies from $2$ to $6$, $c$ varies from $2$ to $6$.There can be other conditions also. Please give the generalized result. It will be really helpful.