Evaluate: $$\int_{0}^{\pi/2}{\frac{dx}{a+\sin^2(x)}}$$ for $|a|&gt;1$  Note: This integral is to be solved using complex analysis (Calculus of Residues)
I want to show that  $$\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{a + \sin^2(\theta)} = \frac{\pi}{2\sqrt{a(a+1)}}$$  for $a &gt; 0$.  I try several methods   Substitutions to rationalize The famous $U = \tan(\frac{\theta}{2})$ Multiplying by conjugates   and other calculus techniques and still, I can not prove the equality.   Anyone can give me a hint of how should I begin to work this integral?  I'm very ashamed that I can not solve this problem.
Is there a way to determine what version (distribution &amp; kernel version, I suppose) of Linux is running (from the command-line), that works on any Linux system? Proof that in a field $x=0$ is equivalent to $x=-x$ Similarities and differences between correlation and regression If I want to investigate how two continuous variables are linked, what is the difference between calculating the correlation coefficient (Pearson's $r$) versus calculating the (simple linear) regression coefficient?  I see people who, if the regression coefficient is significantly different from zero, talk about the two variables as if they are correlated, which is confusing as it suggests that the two coefficients (correlation, regression) are the same thing.  Having said that, isn't $r$ a measure of the (regression line) slope anyway? I'm confused!! To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. I have a question and got strucked on this.. To find the total no. of six digit numbers that can be formed having property that every succeeding digit is greater than preceding digit. Please guide me to solve this. Thanks in advance. Does Kylo not know the fate of Darth Vader? It seems Kylo has an obsession of Darth Vader and has an interest in the dark side because of it.   Does he not realize that Darth Vader turned good and helped destroy the Empire (Spoilers?)? I feel like Kylo would've seen how much better the Light Side was as it was able to turn Vader from the Dark Side if he had known this. HP Easy Scan - "HP Scanner 3" will damage your computer Show that every large integer has a large prime-power factor  That is, if $P(n)$ designates the largest number $p^a$ which divides $n$, then $\lim_{n\to\infty}P(n)=\infty.$ What is the usage of Zero Ohm & MiliOhm Resistor? I am new to PCB design and I noticed that some schematics use 0Ω or 100mΩ resistors. What is their purpose and why do we need to use them in our PCB design?  Normally if we wish to probe how much current the load is taking, we put a jumper pin across the PCB trace (then measure current across the pin using a multimeter). Adding resistors for this purpose seems like it would waste a lot of PCB real-estate. Is this the sole reason why 100mΩ resistors are being placed (since I = V/0.1Ω) instead of a jumper pin?   If so, is there any consideration we should take when placing such a mΩ resistor on board so it doesn't affect the signal or behaviour of the circuit? How did Lucius Malfoy know Sirius's Animagus form? In Order of the Phoenix, Snape eggs on Sirius at Grimmauld Place:  ‘Tell me, how is Lucius Malfoy these days? I expect he’s delighted his lapdog’s working at Hogwarts, isn’t he?’ [Sirius] ‘Speaking of dogs,’ said Snape softly, ‘did you know that Lucius Malfoy recognised you last time you risked a little jaunt outside? Clever idea, Black, getting yourself seen on a safe station platform ... gave you a cast-iron excuse not to leave your hidey-hole in future, didn’t it?’ Order of the Phoenix - Chapter twenty-four, Occlumency - Page 460 - Bloomsbury  How did Lucius Malfoy know that Sirius was a dog Animagus? How did he know it was Sirius in the train station? Some facts to consider: According to the 's timeline, the characters who are considered to be of the Marauders era (born 1959) were only five years younger than Lucius Malfoy (born 1954), so Sirius and Snape, and the other Marauders-era students, and Lucius Malfoy, would have been at Hogwarts together for two years. Prisoner of Azkaban demonstrates it took Sirius, James, and Peter four to five years to master the Animagus transformation, which would have been at least two years after Lucius Malfoy would have left Hogwarts (Although we don't know when Lucius became a school governor.). A canon-based answer would be great; barring one, a speculative answer in the spirit of canon is welcome. What is the easiest way to resolve apt-get BADSIG GPG errors? I frequently cross this issue, and always have to google for an answer. Does anyone have a permanent fix for BADSIG errors from apt-get?     W: GPG error:    lucid   Release: The following signatures were   invalid: BADSIG 54422A4B98AB5139   Oracle Corporation (VirtualBox archive   signing key) says:     A humanoid killed by this spell rises at the start of your next turn as a zombie that is permanently under your command   If a player is hit by finger of death and dropped to 0hp, do they turn into a zombie immediately? If they subsequently fail death saves and die, do they then turn into a zombie? I have recently purchased a hybrid bike with suspension (not sure if that is relevant to this question). It came with two pieces of plastic. Any ideas what they may be for? Which 1970s UK fantasy kids TV show has an older brother die? How to detect if JavaScript is disabled? There was a post this morning asking about how many people disable JavaScript. Then I began to wonder what techniques might be used to determine if the user has it disabled.   Does anyone know of some short/simple ways to detect if JavaScript is disabled? My intention is to give a warning that the site is not able to function properly without the browser having JS enabled.   Eventually I would want to redirect them to content that is able to work in the absence of JS, but I need this detection as a placeholder to start. What is the best way to animate the camera path? I'm modeling a car driving along a road(speeds up, slows down, and stops at traffic lights to wait till the lights turn green, etc.) What is the best way to animate the camera path for the following scenario?   The camera starts moving from point A, and it takes 500 frames to get to point B. During this 500 frames, the camera first gradually speeds up, then moves at a constant speed, and finally slows down to a complete stop at point B. The next 50 frames, the camera will stay at point B. From frame 551, it takes another 500 frames to get to point C from point B. The camera experiences the same motion as described in #1 (speeds up, constant speed, slows down, then stop.) Again, the camera will stay at point C for another 50 frames.  Point C to D is a curve path ( added using Shift + A -> add path). I want the camera to move along the path from frame 1101 to frame 1200. The camera should have the same motion as before(speeds up, constant speed, slows down, then stop.)   What is the best way to animate the camera path in this case? Is there any Python scripts available and can be applied to case like this? Thank you all in advance for your replies. Fourier transform of $\exp(-t^2)$ using contour integration. When one hears that something smells, one would generally assume that it smells bad.   Isn't there a word which wouldn't bring to mind the idea of a bad odor? For example, how would you describe pot-pourri (assuming you like the smell of pot-pourri)? If possible, I would like an adjective, or a word that could takes the place of an adjective. I want to search for what += means, but Google won't return any documents.  Also when I come across some PHP code I don't understand I can't search for the keywords as written.  How can I search for these special characters or include these special characters in a search. If you want to buy a house but you also think value will drop… is there something you can do to alleviate the risk? Disclaimer: yours won't be fiscal advice. People involved will consult a professional before doing anything. This is just to gather some preliminary information.  Situation:   you want to buy a house or a flat, in UK you will get a mortgage in GBP, but you'd pay the down payment in EUR you think that there are fairly high chances that the value of houses in UK and GBP will drop the whole point of this question is actually buying the house/flat, renting is not an option by definition   What can you do to mitigate that risk? Simple newbie help page about avoiding "Thanks"