The easiest way of integration. Integration of fractions.

You all come back; you are asked to differentiate, integrate this—sorry, so this is integration. Before we start, I need to introduce something to you so that you can understand what is actually happening. Here, x to the power of three can be read as four over x to the power of three. can be written as four x to the power of minus three, yes, following the law of indices. told that one over, for example, x to the power of two is x to the power of minus two, that if We are going to remove one layer and we are going to introduce negative sign, so this is exactly what I did here, okay? Plus 2x dx. All right, now what do we do next? Sorry, this DX is actually for these two terms; we are going to separate them. These two terms We are going to separate them because we are integrating them individually, so we are going to have integral of 4x times the minor of 3dx okay plus integral of 2x dx okay at this point let's start our integration is proper at this point, what do you do next? We start integration, and for us to integrate For those of you who are hearing about integration for the first time, this is integration, for example. if we have 2x squared, we are not really interested in this two; we are only interested In the power of our X—that is what we are making use of; the power of our X is our point of focus. so we are going to add one to the power of our x, so in this case we are adding one to two divide this by what the new power is; so the new power is now two plus one, so you add one to your power the power of your x or y or t depending on your variable, and you also divide by what The new power, so we are going to apply this to what we have here, so to integrate this now we are going to have 4x to the power of minus three plus one; we have to add one towards the the power of our x and we are going to divide by our new power, which is minus three plus one All right, and then the same thing with this—yes, I know some of you will be like, "There's no power." X doesn't have any power, but X actually has a power now. What is the power of X? The power of X. is one, yeah, even though it's not there, it is one because x to the power of one is still x, so what we actually have here is 2x to the power of one, so to integrate this, remember that the power is our The point of interest is what we are interested in, so it's going to be one it already has one as power, so we are going to add one to the power and we are going to divide by the new power One plus one, then we add c, which is the constant of integration, so what do we have? Now the next thing we need to do is solve what we have here, so this is going to be 4x minus 3 plus 1 is minus 2 over minus 3 plus 1 is minus 2, all right, plus we have 2x to the power of what 2 2x the power of 2 divided by 2 plus c very interesting right so at this point what Do we notice that these two can divide itself? One divides this four to give you what? Two, so you have. 2 times the power of minus 2, all over minus 1, all right, all over minus 1, plus these two, we can See these two, we are left with what x squared plus c. What do we do next? This minus. 1 when it divides what we have here we are going to have minus 2x to the power of minus 2 plus x to the power of 2 plus c. Now this particular solution is correct, okay? If you have watched to to this extent, I think it's time for you to click the follow button, all right, so that you will Don't miss my next video, all right? This particular solution is correct, but assuming this is an Objective question: There is a possibility that this may not be the solution, all right. This is not what they will bring out, so what else do you think this might end up being? We can write this as what minus 2 over x squared. Yes, we can bring this x to the power of minus 2. down by removing the negative sign, yeah, plus x squared plus c, so these two and this one and This one, both of them are the correct solutions. Thank you so much for watching. And I wish to see you next time, bye.