 Hello friends, welcome to Centrum Academy YouTube channel where we bring for you every day one new problem in the art of problem solving So here we are here. We are with yet another problem in the art of problem solving So these students as you can see on your screen right now We have a question where a integration question has been asked, right? But what is different in this question is you actually do not know the function which you have to integrate Because that function has been written in the form of a puzzle for us, right? So this is exactly what we call in mathematics as Functional equation so they have mentioned the function as a functional equation So they are not very explicit about the function rather. It is very implicitly mentioned as a functional equation So first of all if you want to solve such a question, we need to figure out. What is this function actually? Now there are so many text materials. There are so many resource materials to help you find a Functional equation solution, but actually if you see from the core of it, there is no direct method to find it out So you have to try out you have to put some special values of the inputs and try to see Where are you basically heading for while you're finding the function? So in this case what we are going to do is as you can see there is a functional equation on your screen Which I am highlighting with the yellow color. So in this case what I'm going to do I'm going to start by substituting x equal to f of y in this functional equation Okay, so let's put x as f of y. Let's see what happens to our function So what happens I end up getting f of x minus x on the left hand side on the right hand side I'll end up getting f of x plus x Now f y is x so that will become x into x again, which is x squared plus f of x minus 1, right? So this gives us f 0 equal to 2 f of x plus x square minus 1 Right now. What is f 0 here? f 0 is something which we don't know yet So let's call that f 0 as a c because we know for sure it's going to be some constant, right? So let's replace our f 0 with a c so that leaves us with C equal to 2 f of x plus x square minus 1 So what is f of x? f of x is going to be c plus 1 minus x square over 2 All right now without knowing my c I would not be able to know my function and without knowing the function Unfortunately, I will not be able to find this integral So everything is dependent on me finding the value of c because once we are done with the c We know the function completely and once we know the function completely there is no stopping from there on All right, so let's move on. So what I'm going to do next is something which might surprise you So please please please watch this video carefully So what I'm going to do next is I'm going to put I'm going to put my x value So let's put my x value as let's say something called y1 and let's put the f of y value as y2 Okay, so what I'm doing I'm basically trying to tamper the input of the left-hand side So as you can see the the part which I'm circling for you that x I'm going to put as y1 and f of y I'm going to put as a y2. Okay. Let's see. Where does it lead to? So the left-hand side becomes f of y1 minus y2, right? What happens to the right side? What happens to the right side? My right side becomes f of y2 Isn't it? All right, so now Let us put in the value of f of y2 now. See everybody. Please pay attention This expression gives you an idea of the structure of the function, right? So if f of x is c plus 1 minus x square by 2 It's very easy to figure out that f of y2 will be c plus 1 minus y2 square by 2, right? and Not only that f of y1 can also be written in a similar way as C plus 1 minus y2 square by 2 and finally the minus 1 which is there already in the expression Now, let us simplify this even further All right So this 2 and the ones they will happily get cancelled and I will end up getting 2 c on the top I'll end up getting minus y1 square minus y2 square plus 2 y1 y2 whole upon 2, right? In case you have any issues, please do comment in the comment section and let me know if you are not able to understand any Power of it. I'll be happy to shoot another video for the same Okay, so now Let's write this term as y1 minus y2 whole square So each of my viewers would agree to me that I can write this as y1 minus y2 the whole square Isn't it? Okay. So here we are with the last leg of the vacation, right? Now I would expect everybody now to listen this out very very carefully So here if you have put y1 minus y2 as your input, you are getting the function in terms of y1 minus y2 What does this indicate my dear viewers? It indicates that if I put x in place of y1 minus y2, this should ideally give me c minus x square by 2, right? So all I'm doing is I'm changing the input to the function from y1 minus y2 back to x, right? So when I change it back to x everywhere where there is a y1 minus y2 would automatically get replaced by an x So what does it mean finally? So if you see this is Let's say the expression of the function and we only had one expression over here Let's call this as the two so since one and two correspond to the same may I equate it? Okay, so since one and two relate the same function So can I say c plus 1 minus x square by 2 is as good as c minus x square by 2? Yes, or no Yes, or no, so let's now figure out what's the value of c for us So here I can see that c plus 1 by 2 minus x square by 2 is c minus x square by 2 Let's happily cancel out x square by 2 from both the sides So that clearly means c plus 1 is 2c which means c value is a 1 Right now, this is a very happy news for us because once we get the c value We know the function completely so what is the function so that implies my function which was earlier? C minus x square by 2 now becomes 1 minus x square by 2 right great news Let's wind up the problem now by just integrating this function from 0 to 1 So what is the integral of this function from 0 to 1? Let's write it down Integral of this function from 0 to 1 is as good as saying integrating 1 minus x square by 2 that gives us x minus x cube by 6 and When you put the limits of integration it just simplifies to 5 upon 6 So the answer to this question is 5 upon 6 and the right option is Option number c s So dear students as you can see in this particular problem the problem was not in integration The problem was actually identifying the function which you want to integrate So this is where the integrand was expressed like a puzzle for us And it was very important to solve that puzzle in order to solve this question I hope this gave you a good insight on how to deal with functional equations Thank you so much for watching. Stay safe. See you