show that $\int_0^1 (\sqrt[3]{1-x^7}-\sqrt[7]{1-x^3}) \, dx=0$
How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$
Zero morphism does not depend on zero object I want to prove that zero morphism does not depend on zero object. Let a zero object $0 \in \mathcal{C}$.  I supposed that exists another zero object $0' \in \mathcal{C}$, and I considered $0: A \to B$ and also $$A \to 0 \to 0' \to B$$ But I don't know to prove that zero morphism $0: A \to B$ not does not depend on zero object. Can anybody give me a suggestion? You should choose your Display Name when signing up When a user signs up , their username is something along the lines of "user########"  Granted, in the welcome email it does say something along the lines of "Head over to your user account page to chose a display name!", however only a few people actually read that email.  I'm not saying we should restrict people from changing their display name again in the future - Everyone would still be able to do so. It's just that when you sign up you choose an initial display name that will be what your first display name is instead of "user########". How to kill all process with given name? When should I use \input vs. \include? There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of? I'm very confused about tenses.  I have examples in two different situations here. Situation 1: I went to New York two months ago and was talking to someone about it.  A. This is the first time I've been to New York. B. That was the first time I'd been to New York.  Which is correct in a daily conversation?  Situation 2: I'm going to New York next month and talking to someone.  A. This will be the first time I've been to New York. B. This is the first time I've been to New York. C. This will be the first time I go to New York.  Which sentence is appropriate?  According to a grammar article I've read before, I think it says that I can use this sentence in both cases:  This is the first time I've been to New York.  But I'm not sure if this is really so. So I'd like to know which tense I should use in each case. Could you explain it? What is a current replacement of gnome-open command (generic open of files based on the type)?  Before:   gnome-open mydoc.pdf  # opened PDF in default application   Now:  gnome-open The program 'gnome-open' is currently not installed.  You can install it by typing: sudo apt-get install libgnome2-0 Why can't I sort questions by "votes" or "unanswered" without logging in? On the "Questions" page of each Stack Exchange site, there are links at the top to sort the questions by "newest", "frequent", "votes", "active", and "unanswered".  At least that is what it looks like if I am logged in. If I am on a site that I don't have an account for, or if I am not logged in, the only one I see is "newest".  This seems to have been done recently. I could swear I saw the other links, even when I was not logged in, until a few weeks ago, or maybe later than that.  Why was this change made? When I studied Bayesian statistics, a question about the notation of Bayes' Theorem came to my mind. Below is the density function version of Bayes' Theorem, where $y$ is data vector and $\theta$ is the parameter vector:  $$ p(\theta|y)=\frac{p(y|\theta)p(\theta)}{p(y)} $$ The numerator on the right handside can be written as: $$ p(y,\theta) $$ which is the joint probability distribution of $y$ and $\theta$, then Bayes' theorem could be written as: $$ p(\theta|y)=\frac{p(y,\theta)}{p(y)} $$ Furthermore, $$  p(\theta|y)\propto p(y,\theta) $$ Am I right on this? I think it does not look right. Because the posterior is the proportional to the joint density function. But where is the mistake? Is $\sqrt{64}$ considered $8$? or is it $8,-8$? Last year in Pre-Algebra we learned about square roots. I was taught then that $\sqrt{64}=8$ and $\sqrt{100}=10$, which I understood and accepted. I was also taught that $\pm\sqrt{64} = 8,-8$ because both of those numbers squared is 64, which I also get. But this year, with a new school and teacher in a different state, our teacher is telling us that: $\sqrt{64}=8,-8$ and $\pm\sqrt{64}$ also is $8,-8$. The way to get the positive root of something is: $+\sqrt{64}=8$ And these seem to contradict each other. I was always taught that a regular square root returned a positive number and only a positive number, but now my teacher is saying a regular square root gives two numbers, and considering the square root of a number $n$ is defined as $y^2=n$ I see where he is coming from. Upon researching this Wikipedia says:  For example, $4$ and $−4$ are square roots of $16$ because $4^{2} = (−4)^{2} = 16$  And Wolfram MathWorld says:  Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of $9$ are $-3$ and $+3$  But on the other side, Wolfram Alpha, when given &quot;The square root of 9&quot; gives only 3.  So, which is right? Is $\sqrt{64}$ considered $8$? or is it $8,-8$? Is there an implementable (non-theoretical) way to prove if email address belong to real email account? equal size numerator and denominator I am trying to size the numerator and denominator of the below equation so that they are equal in size.  I have not had much luck suggestions are appreciated.  \begin{equation} \sum_{i=1}^{n} \frac{\frac{cf_{n}}{(1+i)^n}} {\displaystyle\sum_{i=1}^{n}\frac{cf_n}{(1+i)^n}} , n_{i} \end{equation} What Is Energy? Where did it come from? The simplistic undergrad explanation aside, I've never really understood what energy really is. I've been told that it's something when converted from one kind of something to another kind, or does some "work", as defined by us, but what is that something?  Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learned from thermodynamics where it goes, but where does it come from?  I know this sounds like something trivially simple, but there is so much going on in the physical universe and I just can't grasp what it is. Maybe it is because I lack the mathematical understanding that I can't grasp the subtle things the universe is doing. Still, I want to understand what it is doing.  How do I get to the point of understanding what it's doing?  (Note: What prompted me to ask this was . I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.) What's the "Bad magic number" ImportError in python, and how do I fix it?  The only thing I can find online suggests this is caused by compiling a .py -> .pyc file and then trying to use it with the wrong version of python. In my case, however, the file seems to import fine some times but not others, and I'm not sure why.  The information python's providing in the traceback isn't particularly helpful (which is why I was asking here...), but here it is in case it helps:  Traceback (most recent call last):   File "run.py", line 7, in &lt;module&gt;     from Normalization import Normalizer Not able to query standard user field 'StartDay' in salesforce I want to query start of day and end of day field from the user object but I am getting error for that refer images for more details Why can't I run two mysqli queries? The second one fails Can I start and stop a tile floor job? I am tiling my bathroom floor and don't soon have a full day to dedicate to the job. Is it ok to lay down a few tiles each day and come back to the job when I have time? I would only lay mortar for the tiles I have time to lay each day.    My concern is that the mortar wouldn't properly "join" to the previously laid sections of tile and cause an inferior result, but I'm not really basing this concern on any facts. That is what I'm hoping to learn with this question. Neutrino project in Tamil-Nadu India In , the Indian government is going to start a neutrino project.  Questions:   Is it harmful for people who stay there?   Will this project affect any living beings? Does it cause any pollution? When I use any variation of English, US international (with dead keys,altGr dead keys or alternative) on my Linux Mint machine I always encounter this behaviour. When I press one of these keys: ' " and then follow them with a 'non-accentable' character like a [ or a b no output comes out at all.    Whereas in Windows US-International it would print [ or b. If I wanted to type this I would have to escape each dead key with a space instead of with any 'non-accentable' character. This is annoying when programming (not really, but I trained with the Windows 'Qwerty International' on typing.io and switching back and forth between the systems is irritating).    Is there any way to change that so it works like in Windows? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you What motivates people to answer questions in Stack Overflow? I want that too - motivation to answer programming-related questions. What is yours?