integral of sin(x)/x from 0 to inf by Feynman's Technique

[Music] Oh Oh [Music] [Applause] [Music] take a look of this you cover integral from 0 to infinity sex over X in the X world first let me just Erica stop this right here actually converges and second this is rather famous because none of the usual techniques that we know integrations such as integration by parts use up or partial fraction number of them will work for this right here sir I want to give a shout out to security because he wasn't who showed me how to do this famous integral alright and he has this really cool math page I will link the side to learning my description you can so check it out ok and once again Zachary is the man will show me how to do this and maybe you can tell us why this right here converges in the first place and now let's take a look at how this is going to work and this thing is called the final technique it's also called differentiation under the integral sign ok so let's take a look of sex over X the problem for this is that we'll have this X on the denominator if I don't have the X on the denominator of cortex into cosine X that would be nice right how can I get rid of the X in the denominator well how about if I can somehow multiplied by X I'm not talking about multiply the top and bottom by X I'm just talking about multiply by X so that we see X and X can cancel out right don't be so cool isn't it but of course I can I just do the other words I'll change the whole question but let me just write down the function parkin right here let me just put out sine x over X like that and this is how the final technique works all right so we are going to introduce what we call a new parameter and traditionally we use B by course you can use whichever letter that you would like right and let me put down a B right here so I'm considering the function as sine of BX over X if this is the function at half without to multiply by X how can how can I produce that attracts let's take a look at this how about us to be partial derivative with respect to B how's that because I know if I'm looking at this in the B world differentiate at the bottom takes to say and to differentiate sine B X the derivative sine is cosine and the includes a system which is BX now here's the chain rule part we have to multiply by the derivative BX in the B world X is the constant so we will have to multiply by X now and as you can see this X and X 4 10 so and we're just left with cosine of BX and that's wonderful however here is a problem originally we have this sine of X and this is the improper integral going from 0 to infinity so it's somehow I changed the Inca grad I haven't agree anything yet you remember as you have to integrate this on how fast you go to infinity isn't it and of course if you have to concern however this is not convergent anymore unfortunately now you guys have the idea that if I can somehow multiply by X by introducing a new parameter somehow put it next to an X with a function that's how we can multiply by this extra X so that this X and that will cancel each other out that'd be great but this right here didn't work out nicely ok let's go back and see what kind of things that we can possibly do to make this actually work out nicely ok so let me just consider this right here first we have sine X ok Oprah X like that and I'm not going to put on the B divide here anymore because I know that it's not going to work so image is actually like this okay so now I would like to think about maybe I has multiplied by some function and I will insert a be next to the input X so that when I do the partial derivative to the earlier I can produce the extra X so that directs and that will cancel out and now just ask yourself what are the functions that we know and that will work with sigh X in terms of integration damages give us an example would you like two x ln x and of course that says that i'm going to have the beer idea so that if i differentiate this with respect to be i can multiply by the x right well imagine the x is no longer there you have sex and ln x + bx technically can we integrate that no unfortunately right when I say integrate what to make sure to integrate that in the eggs world as well but no well when we have sex I know a function now we can actually integrate with sex namely e to the something right so right here I would like to multiply this function by into the eggs let me just put this down for now right and I'm going to keep track of what exactly I'm doing I have to differentiate sometimes with respect to B sometimes respect to whatever right so as you know I'm going to put a bead right here so that can produce the eggs and the eggs and eggs will cancel now this is my new function I will call this to be I of B B is my input now alright B is my variable in this world how in the world and don't forget we're still talking about integration so we still have to integrate our zero to infinity and this is technically still in the X world so in fact defined IRB to be this integral and now B is the function if you have studied Laplace transform it's kind of similar right because you have the L or fiber and then the in process and you have that into divisor as T right negative as T as is the new variable that you end up with but anyway do you think this will work not quite because if you look at the right hand side in the X world especially this is not going to converge why - just e to the D X of course that goes out to infinity that dominates that parts but this is actually an easy fix because all I need to do is multiply by negative right multiplied by a negative right here so that you know this actually will go down to the denominator and when the egg when the e to the negative B X for this moves to denominator for an exponential part is in the denominator you can almost guarantee it will be a convergent right now will be great okay so this is my definition of I of B well now this right here is to compute this and that what's the difference I have this new part but I can pretend nothing happened because I can say all I need to do is this is the same as a integral from 0 to infinity sine x over x times e to the negative 0 x how's that because of course we know this is just 1 right that's just 1 and now you know I put you right here for one thing is I just want to cutting I oh I just have to plug in 0 into I right so plugging 2 into B so I get out 0 so the original question becomes that which is the same as a I just need to figure out what I of 0 okay however the problem is that I cannot just plug in 0 right now otherwise it defeats the whole purpose because as will be just looking at that right here is that part I'm going to differentiate integrate and things like that it's kind of like solving an exact differential equation if you have studied is already because you have to differentiate and then go back to integrate all that right really cool first we are going to be integrating this right here with respect to B so let me just put that down right here for you guys this is the advantage of I of B I what differentiate well technically on the left hand side I had to put down the regular derivative so from D D be alright because the left hand side I just have to be and I will also do the same because technically only lessons are not writing on the right hand side this right here at yet is a function of B so if you do that you'll get a function of P I will do the complete right here okay okay now let's use a different notation on left hand side GDP of Iove I will use the prime notation or derivative so I'm talking about I PI of B and now here is the cool part I have integral inside and a differentiate the result that integral but I don't want that I want to make the differentiation happen first so that I can have the extra X so that X an actual turns out I'm going to bring this derivative into the integral sign and this is that part for the differentiation under the integral sign and what we are going to do is what we are going to use B like this rule right so I'm going to switch that now we will have the interview on our side for 0 to infinity when I bring the derivative inside look at the inside right here we have two variables technically name DDX and the B so when we bring that inside we have to talk about partial derivative so I'll change notation to the partial with respect to B like this okay and right here I will just write it down sine X and let me put this down on top as well e to the negative B X over X like that right and then this is DX of course okay this is we have from this step to this step this is called like this is the other guy besides Newton right for the calculus like this rule and the whole process called the final technique alright anyway that's what this inside out you know we have so on the left hand side we have AI prime of B and on the right hand side we still have the integral sign of course from 0 to infinity and not check this out the fraction bar stays the same the eggs stay on the bottom because we are differentiating with respect to B write the partial derivative scientists are constantly States so let's put that down the Stags and now e to the negative BX states are saying right but we have to multiply by the derivative of negative e^x in the B world name the X is the constant so we have to multiply by what now you could keep X so we'll multiply by X DX like this right and this is still DX and now check this out X and X cancel not for we have right okay let's take a look we have this negative let me bring that outside so we have the negative in the front and this is integral from 0 to infinity the inside here we just have sine x times e to the negative PI X DX all right and on the left hand side which we have I prying or beat like that okay I can fit a little bit more I guess I will i prime of B this is equal to the negative is still in the front but I'm just going to put down the result integration right here for you guys all right because this strategy that's actually doable because I have to integrate in the X world B will be considered to be a constant now right let me tell you the answer for this and if you want to know how to do this right here from scratch you can check out the video in the description as well ok we need to use integration by parts and the result is going to be just for this pocket you will get negative e to the negative be x times cosine X plus B side over I can put all over B square plus 1 like this ok and we're not going to put on plus C because you know this is going from 0 to infinity and that was in the X world so just DK x go for 0 to X infinity okay of course this negative and a negative cancel to be positive now that's what we have and now how that story continuation arrow which also looks like an inter cosine that's so cool on the left hand side we have I crown of B so let's put that down and on the right hand side I'm just going to select the following technically how to plug in infinity into all the X well you know the terminating part is this e to the negative px right this is a dominating part because this is technically at most one like what's at most one so it doesn't really matter for this is it's just a most two or it technically not but it doesn't really matter at all at most a constant right but this right here goes down to a denominator when X goes to infinity and of course we are talking about B is a path with value if B is not positive then this right here wouldn't converge in the first place because that take a look right here we have e to this negative already and suppose B is also negative then we are talking about negative times negative which is positive and when X goes to infinity that you know we will have e to the past the infinity power and that will make the whole thing infinity right so we must have the condition that bead right here has to be greater than or equal to zero okay in order for this to make sense anyway just trust me that this structure is going to go to zero point X goes to infinity all right and we had to subtract when X is the 0 but anyway let me just write this down on this right here I would say this is 0 this is when X goes to infinity all right and then we have the minus when X is equal to 0 I just have to plug in 2 into audio X so we have e to the negative let me just write it down just to show you guys I'm actually doing some work negative B times 0 like this times cosine of 0 and plus B times sine of 0 and this is over over B squared plus 1 right ok i prime of B this is 0 this is negative and this is 1 right this is 0 this is 1 so 1 negative 1 on the top so well negative 1 over B squared negative 1 over B squared plus 1 right negative 1 negative 1 over B squared plus 1 okay yo cassava now this is not a function I of B anymore this is the derivative of I of B and remember we differentiate it with respect beat with respect to B thirst to get back to the original is B we have to integrate now we're to integrate both sides with respect to what B so I will have to do DB DB and both sides and then xi I will just have I of B and on the right hand side to each cavity yes the integral negative one over B squared plus works in the B world it's just going to be negative in first tangent and input speeds are put this down right here now you see that we have no number no number here so be sure that because this is an indefinite integral we have to put down a plus C right and now with this we are almost done because remember we are just pretty much looking for I of 0 and I just have to plug in because you are here B equal to 0 right here but the problem is I don't know what this DS this is a closed for for I of B I just read on overseas so we have to go back to the board to see how else can we make a connection with this to sort of a see once again this is the closed form for I'll be back remember we had our first original definition file P which is right here right originally we defined it I of B to be the integral from 0 to infinity sine x over x times e to the negative P X DX right well this and that must have a connection so I just have to look at this and we'll say set this equal to that integral from 0 to infinity and we will have the sine x over x times e to the negative be X DX right and now look at right here have the be here for our the B of course here we have the B now I can just choose some people you're plugging plugging plugging hopefully there's a good connection Chacho zero no because when natural 0 this will be 0 and then we have C right here and negative zero times X and equally at zero power we have 1 so this is pretty much just the original yet in another word this is going to be B when B zero zero has killed on overseas so I'm going to choose another number beside zero okay far the numbers can we choose though because this right here is a trouble remember earlier at this page after we integrated I had to plug in X equal to zero right and you see when I plug in X equal to zero in here this exponential part is a dominating part right e to the negative B X X goes to infinity and this will make the whole thing equal to zero Thanks coat hangers doesn't matter okay well why don't we do the same right here this time let B goes to infinity all right so right here let me just write it down I am going to let B goes to infinity I'm going to reduce down for you guys as solo right here will first have negative inverse tangent and let me just put infinity nice I hope you get don't - notation all right and then plus C and this has to be the integral from 0 to infinity and we'll have sine x over X and we will have multiplied this by e to the negative B goes to infinity let me just put infinity right here okay and then the X next to it and then the DX can we figure this out negative interest engine of infinity yes because this is PI over 2 and which is happening PI over 2 right so this is negative PI over 2 and then plus C okay what's this army right-hand side remember e to the negative infinity 4 times X and once again X is just partly because you are talking about plus the X equal to 0 to positive x goes to infinity right excuse positive right actually part of the X goes to 0 plus five anyway this part is pretty much just going to be e to the negative infinity you bring that down to the denominator that will kill the whole inter quantity 0 so this whole thing right here thanks to this part okay it will be just 0 and now when you integrate 0 you still get 0 there's no area it doesn't really matter what this is anymore so owing all this part is just 0 okay now negative PI over 2 plus C equals 0 we know that C will be positive PI over 2 aha so finally we know that I of B is equal to negative inverse tangent of B plus PI over 2 for the C and remember we are looking for I of 0 so that's this integral right now let me just read it down properly for you guys I of 0 it will be negative inverse tangent of 0 and M plus PI over 2 of course this is just going to be 0 so we have just PI over 2 finally let me write this down legitimately because this is such a famous question and once again pipe pops up out of nowhere right so let me write this down ladies and gentlemen the integral from 0 to infinity sine x over X normal III fiber because I'm looking for this which is the same as I of 0 I will close that with DX this right here is just positive PI over 2 ok this is the final result ok this is so much fun isn't it a lot of cool things that just happened right here and once again thanks to Zach rly and he was the one who showed me this part right here so be sure you guys check out the page he has a lot of you messed up over there and hopefully gets like my video as well share this video subscribe thank you so much and I [Music] [Applause] [Music]

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Keywords: integral of sin(x)/x from 0 to inf, Feynman's Technique, differentiation under the integral sign, uncommon integral technique, interesting integral, calculus extra credit problems, DIRICHLET INTEGRAL, Dirichlet integral, feynman's integral technique, improper integral, calculus 2 integral, integral, feynman technique of integration, feynman's technique, integral of sin(x)/x

Id: s1zhYD4x6mY

Length: 22min 44sec (1364 seconds)

Published: Sun Aug 20 2017