if $x &gt; 0$, show that  $$f(x)=\int_0^\infty \frac{\cos(xt)}{1+t^2} \, dt= \frac{\pi}{2}e^{-x}$$  The problem appears in the book , under the chapter derivative and integral of Laplace transformation. So I tried to apply the equation   $$\int_0^\infty \frac {f(t)} t \, dt = \int_0^\infty F(p)\,dp,$$  where $F(p) = L[f(x)](p)$, the Laplace transformation of $f(t)$. However, I failed to manipulate the expression to apply this equality.  I need your help. Please not be misled by my approach, it might be totally wrong.
Let $x \in \mathbb R$  Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ .   Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher was not able to give a valid solution without  resorting to Fourier series.   He actually wanted to prove first that the function is a solution of the ODE $$ y-y'' =0$$ without succeeding in proving it.   Anyway the result sould be $\dfrac{\pi}{2}e^{-x}$  Thanks for any help on this subject.
How to control fan speed? How can I control the computer's fan speed?  On Windows there is a wonderful program called . Remove automatically added keyboard inputs and prevent them from coming back (Windows 10) I have 2 languages installed on my computer, both with a single input method.  I have 2 keyboards: CES-CSQ and ENG-US.    However lately (maybe after the last Win10 update) I started to see 2 additional keyboards in my systray  - namely CES-CS and ENG-CSQ.    Is there there a way to remove those 2 input options (CES-CS, ENG-CSQ) which I do not see in the Control panel's Language options?  Solution described at  question does not apply to my problem because I do not see input options that I want remove in Control panel's Language options. How can I have 2 different mod folders with the new Minecraft launcher? Can you use a readied action to teleport out of the way of a weapon attack? Let's say that a player uses the ready action and says "If a character aims at me with a weapon, I'll use Teleport to move behind cover."  What happens? Does this work? If not, please cite the rules to explain why not. Pull Down Resistors In my quest to understand electrical engineering, I have stumbled across this tutorial:    I have understood the diagrams until I got to switches. I am not sure how switches work on the breadboard or the diagrams. This is the specific one I am thinking of (this is of a pull down resistor):    The implementation is:    Based on the diagram, what I think is happening is: Power goes to the switch, if the button is up then the circuit is not completed. If the button is pressed then the current takes the path of least resistance to pin2 because it has more pull (100ohm &lt; 10kohm).  The way it is described in the tutorial sounds like when the button is up, the circuit is still complete, but the 10k ohm resistor pulls the power to the ground. I am not positive how or why if both the 10k ohm and 100ohm are receiving equal current, the current would get pulled to the ground through a higher resistance than is open to pin 2. Editing the "Self-Learner" Badge Description Stemming from  the text for "Self-Learner" badge is pretty easy to get confused by.  Current Text     Answered your own question with score of 3 or more    Issue  The person in the above question was confused as they thought they needed to answer their own question, and that the question had to have a score of 3 or more. What it actually means is that they have to answer their own question, and the answer needs to have a score of three or more. The way it is phrased is ambiguous, so why not make it clear?  Suggestion  Edit the text as follows:     Provide an answer with a score of 3 or more to your own question   No more confusion. Please will someone explain what time dilation really is and how it occurs? There are lots of questions and answers going into how to calculate time dilation, but none that give an intuitive feel for how it happens. As seen at:           Update: I'm 101% sure the down votes were due to the lack of free hands.    Update 2: likes me, like me not, like me, likes me not... yeah ! Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$? Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$?  I know there are such functions, but I just can't think of any example. What does the human body use oxygen for besides the final electron acceptor in the electron transport chain? My biology teachers never explained why animals need to breathe oxygen, just that we organisms die if we don't get oxygen for too long. Maybe one of them happened to mention that its used to make ATP. Now in my AP Biology class we finally learned the specifics of how oxygen is used in the  due to its high electronegativity. But I assume this probably isn't the only reason we need oxygen.   What other purposes does the oxygen we take in through respiration serve? Does oxygen deprivation result in death just due to the halting of ATP production, or is there some other reason as well? What percentage of the oxygen we take in through respiration is expelled later through the breath as carbon dioxide? Decidability of a predicate. I have the following problem:  Let the following predicates be: $P1, P2, Q1, Q1 : R \to \{0,1\} $ .  It is given that $P1 \lor P2$ and $Q1 \lor Q2$ are semidecidable and $P2 = \lnot Q2$  What can it be said about $P1 \lor Q1$?  I have searched all over the internet and I simply cannot understand the concept of decidability of a predicate, let alone a logic operation with predicates. Let $A$ be a commutative ring and $S$ a multiplicative closed subset of $A$. If $A$ is a PID, show that $S^{-1}A$ is a PID.   I've taken an ideal $I$ of $S^{-1}A$ and I've tried to see that is generated by one element; the ideal $I$ has the form $S^{-1}J$ with $J$ an ideal of $A$. $J$ is generated by one element but I can't see why $I$ has to be generated by one element, maybe I'm wrong. i want to add watermark on some videos but problem is because image in some videos its too small and in some videos its too big, (i think it depends on input link resulution)   so all my videos have output '640:360' so how can i make watermark image to be same in all videos not on some videos smaller in some bigger   ffmpeg -i "link" -i "/var/www/logo/logo1.png" -filter_complex 'overlay=10:10' -vcodec h264 -crf 25 -preset superfast -maxrate 600k -bufsize 600k -aspect '640:360' -s '640:360' .... Search among posts you've voted for Related posts:    These posts are different from mine in that they merely ask for the &quot;votes&quot; tab back or ask if it is at all possible to get such information, while I propose how to implement this as a search feature. Background I recently found myself facing a problem very similar to one that I had solved before. I remembered that I found the solution on SO and I remembered upvoting the post containing the solution. I wasn't able to find that particular post by just searching with relevant keywords. It would have been nice if I could search only among the questions I have voted on. I searched on meta and found that there had been a tab for looking at what you've voted on, but that it was removed due to lack of interest. This is understandable, since just listing the hundreds/thousands of posts you've voted on isn't particularly useful. Proposal What I want is to be able to search for posts I've voted on similarly to the way I'm able to search on my posts within a specific tag. For instance: user:310121 [iphone] apple  This will return all my posts tagged iPhone, where I mention Apple. This is brilliant. This is just how I want search to work. Now what I want to do is to extend this to include votes. The exact syntax isn't relevant, but I had to make one up to show you what I mean: To show all posts you've voted on: user:310121 {votes}  To show all posts you've upvoted: user:310121 {upvotes}  To show all posts you've downvoted: user:310121 {downvotes}  You could probably omit the the user part, since you're only allowed to see your own votes. (Unless it's desirable to be able to look at other peoples votes. Maybe you should be allowed to do this if you have enough rep? Well, that's another discussion.) I guess that even if you're allowed to see other peoples votes it would make sense to display your own votes if the user part is omitted. So in my case I could have done the following search and found the post I was looking for: {upvotes} keyword  Lastly I just want to suggest that clicking on your vote count will work just like clicking on a tag. If I click on a tag in my profile the following search will be done: user:310121 [tag]  So if I click on my upvote count the following search should be made: user:310121 {upvotes}  The same goes for downvotes: user:310121 {downvotes}  This would be a very useful feature and it wouldn't be intrusive in any way. The users that don't care for this feature might never know that it's there. And just for the record the favorite system doesn't replace this feature. How can one change an entire document to use bold font? I see many posts in how to changes the font and font size, but none for changing the default font for a document to bold font. I believe this is called changing the font series. The approach in  (obtained by random Googling) looks like it should work, but doesn't do anything when I stick it in the preamble. Maybe I'm putting it in the wrong place.  \fontencoding{T1} \fontfamily{garamond} \fontseries{m} \fontshape{it} \fontsize{12}{15} \selectfont   There is also some discussion . Intuitively, why does (B → S) ≡ (¬B ∨ S)? I pursue only intuition; please do not answer with  or . I comprehend the following explanation, but it still does not supply the intuition and I sense a deeper, directer, more intuitive explanation. So how else can I intuit this? Or have I neglected something below? I rewrote: p 51-52.  (2010 2 ed) by Henle, Garfield, Tymoczko.    If the kid misbehaves and the teacher forces him to stay, then the teacher spoke truthfully. Suppose the kid misbehaves but travels. Then clearly the teacher lied. But what if the kid misbehaves, but a big snowstorm forces the trip’s cancellation? You can’t say the teacher lied. The kid behaved but didn’t go; nobody went. The teacher spoke truthfully. 1. If the kid behaved and travels, the teacher spoke truthfully.  Notice that the pattern of Ts and Fs we get, is the pattern for disjunction (see p. 38). I remember that the main character is trying to find the group that destroyed his home planet while he was on a mission. His planet was very difficult to live on and the people became expert in martial arts and were the best soldiers in the galaxy. They were mercenaries and often helped against tyrants. There was a very memorable section in at least one book about an initiation task that the main character had to do in that he had to survive in their wilderness for days. He was at some point surrounded in a old building by thousands of poisonous worm/snake creatures that he managed to survive until rescued by the leaders. Please show changed titles separately in edit diffs In an edit diff, when the revision changes the title, removed words are shown struck out in red and added words are shown with a green background. This is fine when there are only removed words, or only added words, but when there are both, the result is difficult to read.        Getting default accessible fields of… No. Getting accessible fields of an object in a Salesforce object… No. Getting accessible fields of a Salesforce object using REST API. Phew. Yeah, ok.        Program causes run. Time error? Oh, Program causes run-time error … when run on specific machine. Yeah, sounds sensible. What was the old title? Oh dear…  Please show both the old title and the new title, one below the other. Check if a class is derived from a generic class I have a generic class in my project with derived classes.  public class GenericClass&lt;T&gt; : GenericInterface&lt;T&gt; { }  public class Test : GenericClass&lt;SomeType&gt; { }   Is there any way to find out if a Type object is derived from GenericClass?  t.IsSubclassOf(typeof(GenericClass&lt;&gt;))   does not work. Can Hydrogen Fusion via CNO Cycle Occur in First Generation Stars CNO cycle requires the presence of carbon, nitrogen and oxygen to undergo hydrogen fusion. Does this mean that for first generation stars, no matter how big they are, can't undergo hydrogen fusion by CNO cycle because there is no carbon, nitrogen and oxygen present?   I can't seem to find any website which mentions this. All they say is if a star is large enough, it will undergo hydrogen fusion via CNO cycle, they don't specifically mention it having to be a second generation star.