How to find the indefinite integral $$\int \frac{dx}{1+x^{n}}$$  where n is a positive integer?
Well, this might be a really simple one. But still... What will be the soln. to --- \begin{aligned} \int\frac{1}{1+x^n} dx \end{aligned}  Is substituting \begin{aligned} 1+x^n \end{aligned} by tan z the correct step?  Thanks for the Help.
How can I animate the Restrict Visibility property of an object? I've been trying to create an animation where an object blinks on and off.   For example, I have a Cube and I set a keyframe on it's Restrict Visibility icon (the "eye" icon) with I, then I toggle off that "eye" so my cube doesn't show in the Viewport and again I press I to record that keyframe. Then, when I preview the animation in the Viewport, I see my cube appears and then disappears like I want, but when I render it, the cube does not disappear at all.  What am I doing wrong? String vs. StringBuilder I understand the difference between String and StringBuilder (StringBuilder being mutable) but is there a large performance difference between the two?   The program I’m working on has a lot of case driven string appends (500+). Is using StringBuilder a better choice? Apache error "Could not reliably determine the server's fully qualified domain name" Skype app in my Ubuntu system work perfectly. But today when I run the app, I see this:    How Can I fix this? I'm looking for specific, real cases in which a causal relationship was inappropriately inferred from evidence of a correlation.   Specifically, I'm interested in examples that meet the following criteria:   Existence of the causal relationship was accepted as fact widely enough to have notable effects (on public policy, discourse, individual decisions, etc.). The link was inferred solely on the basis of correlative evidence (perhaps along with the existence of a coherent but unproven causal mechanism).  Causality has been objectively falsified or at least called into serious doubt.   The two examples that came to mind for me aren't quite ideal:   Sodium intake and blood pressure: As I understand it, it has since been determined that salt intake only increases blood pressure in sodium-sensitive individuals. The existence of a valid causal relationship (although not quite the one that was originally accepted) make this example less compelling. Vaccines and autism: I may have the background wrong, but I believe this link was surmised on the basis of both correlations and (fraudulent) experimental evidence. This example is weakened by the fact that (fake) direct evidence existed.   Note: I've seen this similar question:    My question differs primarily in that it focuses on notable, real-world examples and not on examples in which a causal link is clearly absent (e.g., weight and musical skill). NO2 CSV downloaded files shows zero values for consider date range using Google Earth Engine When I am downloading the CSV file, the results show zeros in the CSV file. How can I download exact NO2 values as shown in Chart after running the mentioned below code in Google Earth Engine? // Create a FeatureCollection from the list and print it. var BergamoGeometry = ee.FeatureCollection(BergamoGeometry); print(BergamoGeometry); Map.centerObject(BergamoGeometry); Map.addLayer(BergamoGeometry); //////////////////////////////////////  var dataset = ee.ImageCollection(&quot;COPERNICUS/S5P/NRTI/L3_NO2&quot;)                  .filterBounds(BergamoGeometry)                   .filter(ee.Filter.date('2019-03-30', '2019-04-29'));                   var NO2 = dataset.select('NO2_column_number_density'); var band_viz = {   min: 0,   max: 0.0002,   palette: ['black', 'blue', 'purple', 'cyan', 'green', 'yellow', 'red'] }; // Create and print the chart. print(ui.Chart.image.series(NO2, BergamoGeometry, ee.Reducer.mean(), 30)); Why do shields in Star Trek "wear out"? Combinatorics riddle: keys and a safe There are 8 crew members, The leading member wants that only a crew of 5 people or more could open a safe he bought, To each member he gave equal amount of keys, and locked the safe with several locks.  What's the minimal number of locks he should put:   at least 5 crew members would be required to open the safe? Any team of 5 crew members could open the safe, but none of the 4 team crew members.   1,2 stands individually of course. I am trying to write a condition into a SQL WHERE clause that forces the column comparison to a variable to be compared in lowercase. However the addExpression function isn't accomplishing this (since that is putting the expression in the field selection, not the where clause.  Here is what I have tried:  $category = 'mobile wifi'; $query = db_select('taxonomy_term_data', 'ttd')     -&gt;fields('ttd', array('tid')); $query-&gt;innerJoin('taxonomy_vocabulary', 'tv', 'ttd.vid = tv.vid'); $query-&gt;addExpression("LOWER(ttd.name) = $category"); $result = $query-&gt;condition('machine_name', 'images_cat', '=')     -&gt;execute()     -&gt;fetchAssoc();   And this:  $category = 'mobile wifi'; $query = db_select('taxonomy_term_data', 'ttd')     -&gt;fields('ttd', array('tid')); $query-&gt;innerJoin('taxonomy_vocabulary', 'tv', 'ttd.vid = tv.vid'); $result = $query-&gt;condition('machine_name', 'images_cat', '=')     -&gt;condition('LOWER(ttd.name)', $category, '=')     -&gt;execute()     -&gt;fetchAssoc();   The second one becomes invalid and the first one causes the problem I mentioned above. Any thoughts or suggestions?  Thanks, Patrick I have a homework question that asks     Find necessary and sufficient conditions on $1 \leq i &lt; j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.   Here is what I have done so far.  Call $c = (1 \, 2 \, \dotsc \, n)$.  I made the observation that $$ \overbrace{c c \dotsb c}^{n - j + 1} (i \, j) \overbrace{c^{-1} c^{-1} \dotsb c^{-1}}^{n - j + 1} = (1 + i - j + n \, 1). $$ Thus, $\langle (i \, j), c \rangle = \langle (1 \, i - j + n + 1), c \rangle$, so it suffices to look at $\langle (1 \, k), c \rangle$ for positive integers $2 \leq k \leq n$.  I've checked that, for the value $k=2$, $\langle (1 \, 2), c \rangle = S_n$.  This immediately implies $\langle (1 \, n), c \rangle = S_n$.  I suspect that no other values of $k$ will work, but am not sure how to prove it. Kuratowski-Zorn Lemma with pre-order (quasi-order, proset) instead poset? What would happen if we would use pre-order (there is no weak-antisymmetry, only reflexivity and transitivity) in  Kuratowski-Zorn Lemma instead of partial order?      Suppose that we have set P which is preordered and  has the property   that every chain (i.e. totally ordered subset) has an upper bound in   P. Then the set P contains at least one maximal element.   Is this statement true in ZFC? If not, what can we postulate to make it true? I think it can be false, or at least we need to postulate something more (or not?), but I am just missing something and don't see it.  P.S.      I wanted to put tags with combination of words like: Kuratowski Zorn Lemma Pre-order Proset, but it seems that none of them exist and I need to have 300+ reputation to create new ones, thus I only typed (set-theory) :( If someone with bigger reputation could make this topics' tags more precise, I would be grateful :) Simulate repeated rolls of a 7-sided die with a 6-sided die What is the most efficient way to simulate a 7-sided die with a 6-sided die? I've put some thought into it but I'm not sure I get somewhere specifically.  To create a 7-sided die we can use a rejection technique. 3-bits give uniform 1-8 and we need uniform 1-7 which means that we have to reject 1/8 i.e. 12.5% rejection probability.  To create $n * 7$-sided die rolls we need $\lceil log_2(  7^n ) \rceil$ bits. This means that our rejection probability is $p_r(n)=1-\frac{7^n}{2^{\lceil log_2(  7^n ) \rceil}}$.  It turns out that the rejection probability varies wildly but for $n=26$ we get $p_r(26) = 1 - \frac{7^{26}}{2^{\lceil log_2(7^{26}) \rceil}} = 1-\frac{7^{26}}{2^{73}} \approx 0.6\%$ rejection probability which is quite good. This means that we can generate with good odds 26 7-die rolls out of 73 bits.  Similarly, if we throw a fair die $n$ times we get number from $0...(6^n-1)$ which gives us $\lfloor log_2(6^{n}) \rfloor$ bits by rejecting everything which is above $2^{\lfloor log_2(6^{n}) \rfloor}$. Consequently the rejection probability is $p_r(n)=1-\frac{2^{\lfloor log_2(  6^{n} ) \rfloor}}{6^n}$.  Again this varies wildly but for $n = 53$, we get $p_r(53) = 1-\frac{2^{137}}{6^{53}} \approx 0.2\%$ which is excellent. As a result, we can roll the 6-face die 53 times and get ~137 bits.  This means that we get about $\frac{137}{53} * \frac{26}{73} = 0.9207$ 7-face die rolls out of 6-face die rolls which is close to the optimum $\frac{log 7}{log6} = 0.9208$.  Is there a way to get the optimum? Is there an way to find those $n$ numbers as above that minimize errors? Is there relevant theory I could have a look at?  P.S. Relevant python expressions:  min([ (i, round(1000*(1-(  7**i )  /  (2**ceil(log(7**i,2)))) )/10)  for i in xrange(1,100)], key=lambda x: x[1]) min([ (i, round(1000*(1- ((2**floor(log(6**i,2))) / (  6**i ))   )    )/10)  for i in xrange(1,100)], key=lambda x: x[1])   P.S.2 Thanks to @Erick Wong for helping me get the question right with his great comments.  Related question: I accidentally posted a question without being logged in. Is there a way to associate that question with my account?   For more information, visit &quot;&quot; in the .   a.k.a. How can you link a registered account to a cookie-based account? a.k.a. How do I merge an unregistered account with a registered account a.k.a. Merging users a.k.a. How do I associate anonymous questions I’ve asked with my registered account? a.k.a. Is there a way to claim an unregistered user? (in case anyone searches with those terms) How to convert all shapefiles in folder into KML using ArcPy? I was trying to convert all the shp in a folder into kml.  featureclasses = arcpy.ListFeatureClasses() for fc in featureclasses:  # Set Local Variables   composite = 'COMPOSITE'   pixels = 1024   dpi = 96   clamped = 'CLAMPED_TO_GROUND'   scale = 1   outKML = fc[:-4] + &quot;.kmz&quot;   arcpy.LayerToKML_conversion(fc,outKML, scale, composite,'', pixels, dpi, clamped)   It always says Failed to execute. Parameters are not valid. ERROR 000732: Layer: Dataset ZZZ.shp does not exist or is not supported Failed to execute (LayerToKML). But I can manually do it within ArcMap 10.1 Desktop... Find children of the process How to prove that $\lim\limits_{x\to\infty} f(x)/x=L$ Let $f\colon(0,\infty)\to\mathbb R$ be differentiable on $(0,\infty)$ and suppose that $\lim\limits_{x\to\infty} f'(x)=L$.         a. Show that for any $h&gt;0$, $\lim\limits_{x\to\infty} \frac{f(x+h)-f(x)}h=L$.      b. Show that $\lim\limits_{x\to\infty} \frac{f(x)} {x} =L$.   this is a differentiation question.  i don't know how to solve this question  Because it is Korean, it is difficult to deal with this cite. So for me, it is not easy to write.  thank you What is the order when doing $x^{y^z}$ and why? Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why?   Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$?  I always get confused with this and I don't understand the underlying rule. Any help would be appreciated! Can you launch Remote Desktop via URL? Why can other users see the files in my home folder? I just added a new, underprivileged "desktop user," and I was surprised to discover that it can see the files in my home folder.  What is the rational for setting up such lax permissions? Finite groups with exactly one maximal subgroup