I'm trying to understand what's happened . From what I've found online it should be $\sin^{-1}$ not $\cos^{-1}$. But then the proof doesn't work. I'm probably just missing something...  I'm assuming that the equation in the picture isn't of the form suggested in the title or something?
Find the exact value of     $$ \int_{0}^{a} \frac{1}{\sqrt {a^2-x^2}} \mathrm {dx} $$   Where, $a$ is a positive constant  Hi, guys can give me tips to solve this ? Should we use like u substitution?
How many roots have modulus less than $1$? If roots of the equation   $$x^7 - 4x^3 + x + 1=0$$        are plotted on the Argand plane, how many of them have distance from the origin less than $1$?     I found, by plotting the rough curve of $y= x^7 - 4x^3 + x + 1$ that it has three real roots, out of which two of them have modulus greater than $1$ and one has modulus less than $1$. But I don't know how to do the same for non - real roots. The answer given in my book is that $3$ roots have modulus less than $1$. How do "badges" work? How to clone ArrayList and also clone its contents? What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$? /A problem from the 2012 MIT Integration Bee is $$ \int_0^1\frac{x^7-1}{\log(x)}\mathrm dx $$ The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it by hand? Please give me the proof for the following: Let $a_1,\,a_2,\,\dots\,a_n$ be $n$ positive real numbers whose product is equal to $1$. Prove that $$a_1+\cdots+a_n \geq n$$ and that the equality sign holds only if every $a_k$ is  equal to $1$. Spambots causing server crashes. How do I block them? Citing @inbook with biblatex: how to put authors/editors before book title? Sensitivity to initial conditions and topological transitivity Calculate battery life I would like to know how to calculate battery life when using an LED strip. The LED strip is powered by 8 AA batteries in series. LED Current Draw = 260 mA. LED power consumption: 3.1 W. LED Operating Voltage range: 9-14.8 VDC. Please see attached images for battery datasheet.  This is my calculation, but I'm not sure if it's correct:     Battery Life = (Battery Capacity)/(Load Current) = 2000 mAh /260 mA = 7.7 Hours   This is for one battery, and the lights are powered by 8 batteries, so is the total battery life = (7.7)x(8) = 61.6 hours ? Where did the brush size lock go for sculpt mode in 2.8??? How can I block all traffic *except* Tor? How can I define a math operator behaving like \lim or \sum with limits? Viewing all my questions from Android app I want to quantify the relationship between two variables, A and B, using mutual information. The way to compute it is by binning the observations (see example Python code below). However, what factors determines what number of bins is reasonable? I need the computation to be fast so I cannot simply use a lot of bins to be on the safe side.  from sklearn.metrics import mutual_info_score  def calc_MI(x, y, bins):     c_xy = np.histogram2d(x, y, bins)[0]     mi = mutual_info_score(None, None, contingency=c_xy)     return mi Biometric information for a UK visitor visa for the second time I submitted my biometric information when I applied for (and successfully received) the UK visitor visa.  Now, the visa has expired and I would like to apply for it again.   Do I need to submit my biometric information again?  Will the process be the exact same as before? Can I travel to the USA while working remotely for my non-US employer? I have been reading Geoff Cumming's 2008 paper  [~200 citations in Google Scholar] -- and  am confused by one of its central claims. This is one in the series of papers where Cumming argues against $p$-values and in favour of confidence intervals; my question, however, is not about this debate and only concerns one specific claim about $p$-values.   Let me quote from the abstract:     This article shows that, if an initial   experiment results in two-tailed $p= .05$, there is an $80\%$   chance the one-tailed $p$-value from a replication will fall in   the interval $(.00008, .44)$, a $10\%$ chance that $p &lt; .00008$,   and fully a $10\%$ chance that $p &gt; .44$. Remarkably, the   interval—termed a $p$ interval—is this wide however large   the sample size.   Cumming claims that this "$p$ interval", and in fact the whole distribution of $p$-values that one would obtain when replicating the original experiment (with the same fixed sample size), depend only on the original $p$-value $p_\mathrm{obt}$ and do not depend on the true effect size, power, sample size, or anything else:     [...] the probability distribution of $p$ can be derived without   knowing or assuming a value for $\delta$ (or power). [...] We do not assume any prior knowledge   about $\delta$, and we use only the information $M_\mathrm{diff}$ [observed between-group difference] gives about $\delta$ as   the basis for the calculation for a given $p_\mathrm{obt}$ of the distribution of   $p$ and of $p$ intervals.      $\quad\quad\quad$    I am confused by this because to me it seems that the distribution of $p$-values strongly depends on power, whereas the original $p_\mathrm{obt}$ on its own does not give any information about it. It might be that the true effect size is $\delta=0$ and then the distribution is uniform; or maybe the true effect size is huge and then we should expect mostly very small $p$-values. Of course one can start with assuming some prior over possible effect sizes and integrate over it, but Cumming seems to claim that this is not what he is doing.  Question: What exactly is going on here?    Note that this topic is related to this question:  with an excellent answer by @whuber. Cumming has a paper on this topic to:  -- but that one is clear and unproblematic.  I also note that Cumming's claim is repeated several times in the 2015 Nature Methods paper  that some of you might have come across (it already has ~100 citations in Google Scholar):     [...] there will be substantial variation in the $P$ value of repeated experiments. In reality, experiments are rarely repeated; we do not know how different the next $P$ might be. But it is likely that it could be very different. For example, regardless of the statistical power of an experiment, if a single replicate returns a $P$ value of $0.05$, there is an $80\%$ chance that a repeat experiment would return a $P$ value between $0$ and $0.44$ (and a $20\%$ change [sic] that $P$ would be even larger).   (Note, by the way, how, irrespective of whether Cumming's statement is correct or not, Nature Methods paper quotes it inaccurately: according to Cumming, it's only $10\%$ probability above $0.44$. And yes, the paper does say "20% change". Pfff.) Tools to get a pictorial function call graph of code I have a large work space which has many source files of C code. Although I can see the functions called from a function in MS VS2005 using the Object browser, and in MSVC 6.0 also, this only shows functions called from a particular function in a non-graphical kind of display. Additionally, it does not show the function called starting from say main(), and then the functions called from it, and so on, deeper inside to the leaf level function.  I need a tool which will give me a function call graph pictorially with functions callee and caller connected by arrows or something like that, starting from main() to the last level of function, or at least showing a call graph of all functions in one C source file pictorially. It would be great if I could print this graph.  Any good tools to do that (need not be free tools)? I want to add a comma in the thousands place for a number.     String.Format()? What does the Visual Studio "Any CPU" target mean?