How to integrate fractions. The easiest way.

integrating a fraction, I'm going to show you the easiest way you can use to integrate this First of all, we need to understand what integration is. I'm going to tell you. what you really need to do if you want to integrate is to integrate everything you need are interested in the power of your words, variable, for example, the power of t, you will add one to the power of t and you are going to divide by your new power two plus one that is Integration—there is nothing to add about it, so this is exactly what we are going to do here, but Before that, we have a fraction, so how do we now simplify this fraction before we start integrating? That's where this particular topic is going to become very, very interesting, okay? So let's do that. then if you have a fraction and you want to integrate it, I will advise you first of all This fraction can be simplified further to two over three. divided by t squared plus three t squared divided by t squared minus one divided by t squared All right, everything's okay now. We are going to go back to indices. I know that some of you forgotten, but in this case you don't have any options, so let's go back to our indices, what What do indices say about two numbers with the same base that are dividing each other? indices state that if you have a number that is, if they are of the same base, and if you have two numbers that have the same base, for example, we have x to the power of three divided by x to the The law of indices states that we should choose one of the bases, which is x. and we are going to subtract the powers now; when we do that, we are going to have x to the power of one which is also equal to x, just pick one of the bases and subtract the powers; it's easy as that we are going to use this knowledge now to do what to work on this particular question that We have this here, so let us simplify it further. We are going to have the integral of t over three. over t over two, notice that they have the same base, right? And they're dividing each other according According to the law of indices, we are simply going to choose one, so we are going to pick one of the bases. We are going to subtract what the power, okay, three minus two plus, we write three t two minus Two, okay, instead of us stopping ourselves here, we can just cancel this and this, so we are left With only three minus signs, we have what this is the same as t to the power of minus two, yes, let me according to our indices, the rule states that if you have one over x, then it is equal to x to the power of minus one; that is, if you want to eliminate your 1 over x, you introduce your All right, in the same way, if we have one over x squared, it's not going to be x to the power of minus two because the power of x here is two, so similarly, that is exactly what I did; this becomes to the power of minus two. Now at this point, I believe that some of you are just seeing this channel for the first time, so please, I think it's time for you to click on that subscription button and then on your notification bell as well, so that you'll be notified anytime I upload a new one Video, I'm sure you wouldn't want to miss my next video, right? Okay, now let's continue, please. help me to comment, okay? Comment, comment, like, like, share, so that this particular video will be distributed more for the benefit of our students, all right? Okay, let us continue now. This is a noun, simplified, so this same thing as the integral of three minus two gives us one. All right, so t to the power of one, which is still t. All right, because t to the power of one is still t. so t to the power of t plus three minus t to the power of minus two dt now this is what we'll be solving so what Do we do it from here? From here, we need to separate this separately. We need to separate these terms. Yes, because we are going to integrate them separately as well, so it's going to happen. the integral of t dt plus three times the integral of 3 dt minus the integral of what t to the power Minus two, all right. Then this is what, sorry, dt, this is what we are going to integrate now. I did nothing; instead, what I did was separate these terms: integral of t dt integral. of three dt and the integral of what t minus two dt this dt is just telling us that we are integrating With respect to t, it's not actually that we are going to integrate t, but it's merely indicating us that we are integrating with respect to t, so we start our integration even though there is no Turn on this T; we know that the power of T here is one, yes. When there is no power on your the power is one because t to the power of one is still t; now, if we want to integrate this It's going to be t to the power of one plus one, right? In integration, remember what I explained. At the beginning of this particular video, for you to integrate, you add one to the power. of your variable and you divide by the new power; our new power now is one plus one Okay, and the same thing here—we're going to have three, okay? In this case, it seems there is no t. not to mention the power of t that we are going to integrate, but we all know that already to the law of indices, any number to the power of zero is one, so that means that t to the power of zero is one zero is equal to one, so it means that if I add t to the power of zero to multiply this three that which has not changed anything is still what three times one, which is equal to three, therefore because t to the power of zero doesn't change the value of our three, which is this, I'm going to Write it here, I'm going to add it to our variable, to our constant. Sorry, so anytime you have only constant, and you are asked to integrate, add the variable raised to the power of zero; for example, if what What we have here is the x that is, we have t x x x the x, the x. We are going to add x to the power of zero, I added t to the power of zero because we are differentiating with respect to t, remember Okay, so let's go ahead and solve this. If this is what we have now, we're going to add one to the power of our t, and we are going to do or divide by the new power, all right? Minus, this is t to the power of minus two we are going to add one to the power and divide by the new power which is minus two plus one, then at this point we add our constant of integration, this plus c, we need Okay, then let's continue. This is t one plus one, which will give us two over what two plus three. minus t minus two plus one will give us minus one over another minus one plus c, okay, this is equal to t squared over t plus this will still give us 3t, right? Yeah, because t to the power of one is still three, is still t minus, now please pay attention, you know that Minus minus will give us plus. Yes, this divided by negative one will still give us another negative. Sign right so that negative and this negative is going to give us plus, so negative negative. return this to plus then you have t to the power of minus one plus c now this particular solution It is correct 100 percent correct, but if this is an objective question, there is a possibility that You won't see this exactly as it is; there's another way we can express this. the solution is one over two t squared plus three t plus, remember that This has a negative sign, right? To the power of minus one. If we want to remove this negative... sign we'll bring the t down right, so it's going to be one over t plus c; it's supposed to be t to power of one, but t to the power of one is still t, all right, this is the final solution, I am so glad you watched until the end, thank you so much for watching. If you've made it this far, I want to Those who actually watched until the end, I'd like you to comment nines—I really want to know. How many of you watched this video till the end? Just help me by commenting nines in the comments. Section: Let's confuse those who actually didn't watch until the end. Thank you so much for watching. Have a lovely day, bye