$$\int_{0}^1 (1 + 4y^2)^{1/2} dy$$   So, how do I integrate this without the use of trigonometrical substitution? Can anybody give me a hint? Thank you!
How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ?   shows the closed form of it but not the steps in solving it.  I think I need some algebraic trick..   I thought of some hyperbolic functions and their derivatives and equalities:  If I make $ t = \sinh x$, then I can reduce the integral to  $$ \int \cosh ^2 x dx $$ though I'm not sure if this is valid. But from here, I cannot go on evaluating the integral...
Non-perpendicular hyperplane decision trees The usual implementation of decision trees, and estimators based on ensembles of decision trees, use decision trees that threshold on a single predictor at each split. So they divide the feature space into perpendicular hyperlanes.  Is there any benefit to generalizing this a tiny bit. So at each split, the decision tree uses 2 features, and thresholds based on whether $ax_1 + bx_2 \ge T$. In other words, the feature space is now getting divided into hyperplanes that are not necessarily perpendicular.   I think the main disadvantages would be:    increased computation, and  overfitting since now using more degrees of freedom   I guess to help both of these issues, randomization could be used analogous to extra trees. The pair of features could be randomly selected, instead of exhaustively trying every pair, and the threshold T and parameters a and b could be chosen randomly as well (perhaps from a set of few discrete values such that the hyperplanes could have a few orientations).  Are there some cases where you think these non-perpendicular decision trees would be an improvement over the regular implementation of various decision tree algorithms? VirtualBox 5.0.40 crashes the whole host since update to kernel 4.13.0-26-generic mkdir -p will create a directory; it will also make parent directories as needed.  Does a similar command exist for files, that will create a file and parent directories as needed? What's the difference between postfix/smtp and postfix/smtpd A bit confused when reading through the logs of my smtp/mail server, I'm turning this questions to you guys.   In the maillog files I'm seeing entries for postfix/smtp as well as postfix/smtpd. I can also see that there are possible settings for smtp as well as smtpd in the file main.cf  Trying to find some reasonable explanation through google search seemed to be impossible, so here is my question to you guys.  My server was kind of compromised and the spam mails where going through postfix/smtp and not smtpd. Please help me to understand. Thanks in advance! Squares in arithmetic progression It is easy to find 3 squares (of integers) in arithmetic progression. For example, $1^2,5^2,7^2$.  I've been told Fermat proved that there are no progressions of length 4 in the squares. Do you know of a proof of this result?   (Additionally, are there similar results for cubes, 4th powers, etc? If so, what would be a good reference for this type of material?)    Edit, March 30, 2012: The following  in MO is related and may be useful to people interested in the question I posted here. Why is it called the Millennium Falcon? Difference between phrase and idiom What is the difference between a phrase and an idiom? Finding the missing digits of $23!$ It is given $23!=2585201xy38884976640000$. Now it is required to find the value of $x$ and $y$. I know I could find it by using divisibility rules and solving simultaneous equations. Is there any other way to solve it (without computing it by a calculator)? This question is just out of curiosity. Where can the last FSTrim be seen as successful or not? (log files) Referencing bib files with spaces in the filename I am using a Mac with TeX Live and Mendeley for automatically generating .bib files, some of which have spaces in the file names. I would like to know how to reference such files from LaTeX as using "" or \ doesn't seem to work. 60% Downpayment on house? How do I completely remove Ubuntu 12.04 on my Mac? I have dual booted Ubuntu 12.04 on my Mac using disk utility and the rEFIt tool.The battery life is very bad and therefore I want to switch back to Ubuntu 10.04 . How can I completely remove the 12.04 partition? So, the spiffy new wi-fi thermostats want power on the C wire from thermostat, but the old thermostat doesn't have a C wire at it.  What's the easiest way to add the C wire/24 volt to the thermostat location? System settings won't open on Ubuntu 20.04 LTS after adding System76/pop PPA What's the fastest and easiest to read implementation of calculating the sum of digits?  I.e. Given the number: 17463 = 1 + 7 + 4 + 6 + 3 = 21 Finding the unique rock with its weight I have 12 rocks, all have the same weight except for one rock, but I don‘t know which rock is. How do I know which rock is different from other, and also know that rock heavier or lighter, with maximum 3 comparating steps. My only tool is just a balance that can only compare the weight between the rocks. How to make files protected? How to make some file that contains crucial data protected? For instance, suppose I have some files that are important, is there any method or strategy to make these kind of file more protected on Ubuntu? I even want to make some files ask for a password whenever someone tries to open it. Does it affect the version of Ubuntu in terms of security policy? May I use any encryption algorithm externally? Regards. Strange Recurrence: What is it asymptotic to? So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question.... Harmonic Function with linear growth Proving the Lorentz invariance of the Lorentz invariant phase space element I have been looking around for a satisfactory answer to prove that  $$\frac{d^3\vec{p}}{2E_{\vec{p}}}$$  where $E_{\vec{p}}=+\sqrt{(|\vec{p}|c)^2+(mc^2)^2}$, is Lorentz invariant. The standard answer seems to be that the above measure is equal to  $$d^4p \, \delta (p^2-m^2)\theta(p_0)$$  where $p^2=p_0^2-|\vec{p}|^2$, and $\theta(x)$ is the step function. I understand these two are equivalent, but I don't understand why the second has to be Lorentz invariant, in particular why the Dirac delta has to be Lorentz invariant. I have found a  (section 2.1) that proves that $\delta^{(4)}(p-p')$ is Lorentz invariant, but I can't find a way to extend their method successfully here. In fact, all I can seem to get tells me the above isn't Lorentz invariant, and that in fact it should transform to   $$\frac{d^3\vec{p}'}{2\gamma E_{\vec{p}'}}$$  which makes sense from $d^4p$ being Lorentz invaraint and $dp_0$ transforming proportional to $\gamma$, but it isn't what everyone else says. What's the problem here?    An alternative way to "derive" the factor of $1/\gamma$:  $$\delta (p^2-m^2)\theta(p_0)=\frac{1}{E_{\vec p}}\delta(p_0-E_{\vec{p}})$$  Now, as $dp_0$ transforms to $\gamma dp_0$, the $\delta(p_0-E_{\vec{p}})$ should transform to $\gamma^{-1}\delta(p_0-E_{\vec{p}})$, as can be shown with an analogous argument to the one shown in the . The $E_{\vec{p}}$ also transforms to $\gamma(E_{\vec{p}}-v p_x)$ for an $x$-boost, but this doesn't seem to solve it.