Okay, in this video I'll show, you guys how to integrate cosine of X DX. And this right here, it's a pretty standard calculus 2, integral. So if you are taking Cal - then this is for you and I also show you guys two ways to do this. So be sure to check this out be sure you watch both ways and let me know which way you like better. And before we start, let me just ask you guys, please subscribe if you haven't done so already, and I want to thank all. My supporters throughout the years for supporting my.
Channel anyway, the first ways that we are going to start off with you stop that you equal to the inside function. So I'll write down u equals 2l + X, and then I would differentiate both sides. So we get D u equals to 1 of X DX and to isolate the DX let's, just multiply by example, sides. So we see that DX equals 2x the so that's pretty much it.
And now we will take this integral to the new world. And we see this is the integral of cosine. And the input is the right here. And the DX is this, but you. Know if you just put X to you right here, this X is not going to be invited in the new world. And unfortunately, we don't have the over X dumped, a lot to cancel things out so that's, not good. But ok, this is just easy fix to do.
So you look back to you equals to our next. And you know, we can just do e to this power, e to the power. So that yellow and cancel. And this it's a same as saying, e to the u equals 2x. So right here for this X I will actually replace that with e to the U I will put this right here. And I will just write it down e to the U along with this, the U. And now this integral is completely in the new world.
But the question is, how can we continue from here? We need to use integration by parts, and I'll show you guys with a DI format, okay? And hopefully guess have seen my other videos of similar videos on this. If you haven't, you can check out all the links in the description all right. Okay.
So I will just do this on the site because this is a repeated situation. So I will just make a note. On the side let's write this down right here we have to integrate.
And let me just put on e to the first and then cosine, you do you and I will use the DF format. So I will just put on the and an and let's put on some plus minus to get ready. And remember the D high format, it's, not magic. This is just integration by parts.
This is just a much easier way to set up to format, the integration by parts, the D it's like to you the eyes like the TV in the traditional new TV format. But anyway, Enough talking let's, see, which one shall we pick to be integrated and the answer to that is it doesn't matter. So I will just go ahead and do cosine you right here.
And if you wanted to pick each will you right here, that's, okay, too. But anyway, you can try that on your if you reverse this, the answer in the end will be the same anyway. Anyway, since I make my choice, I'll just continue to integrate cosine you. We will actually get past deep sighing you.
And we are doing the integral so be really careful. With the plus minus sign, right? Integral, cosine U is positive sine U. And you can do this again, integrating. Thank you.
We get negative cosine u. So this is pretty much it. And you see that if you differentiate e to the U, you just keep getting each to you, and in fact, I'm going to stop right here because I see the function part, e to the U and I also see the cosine u right here and that's, a repeat of the original I know we do have the miners, but we are paying attention to the function part. If it's a. Repeat you stop. So you didn't need the last row, these three rows enough. And now to continue to remember, whenever you multiply the diagonal along with the sign in front that's, the answer already.
So the first part of the answer is positive, e to the U times sine U so let's write that down e to the sign you. And next I will do this diagonal, but be sure you're attached - in front - e to the U times negative cosine u - - you know it become plus, and we have e to the U times cosine u like this. And.
Lastly, I will take the product of this row. And whenever you do a part of a row, this is still going to be an integral. And you see you, multiply pass the x - you get -.
So this is an - and remember a part of a row it's joining the growth. So that's, why I put on Interpol right here? And because I put a minus out already. So the insides just e to the U times cosine u like this. And then of course, since this is integral so that's attached to D u. And now you see on the left-hand side, we have the integral. E. To the u, cosine u TU.
And the right-hand side we have minus intercrop interview cosine u, TU. So I just have to bring this to the left-hand side, come by them and I can solve for that integral. So I will just add the integral of e to the u. Cosine u Du u on both sides, this way, listen now cancel.
So I will come here to do the same I, add the integral of e to the u. Cosine u Du u. Ok. So we see that this is the wanting to quote any plus another, integral, you get two integrals haha, two integrals of e to the U. Times cosine u TU.
And then the right-hand side states, the same right here. So we have e to the U sine u, plus e to the u, cosine u and I want to isolate this integral, but we have the two in front. So what can we do? We can simply multiply by one half right, and I'll just multiply by one half on both sides so that this.
And now we can solve it. And on the right-hand side, I just have the integral e to the U cosine u, the U being equal to. Of course, you distribute of one half. So you get one half e to the U. Sine u one half, right? You add right here and e to the U cosine u and that's pretty much. If I'm, not going to put on plus C yet, because this is like the appetizer round.
Three is right here. Right. Anyway. This right here is just going to be that. So let me just write this down. We have one half e to the U sine U.
So let me just put this down. And then we add it with one half e to the u cosine u, right? So that's pretty much it for that. Will we do the integration? Now we just have to go back to the X world. So we see one half is the other one half e to the U is X.
So I, just put that down right here and sine u is Ln X. So right here I will write down sign of Ln X. And then we pretty much do the same for the rest. We add the words one-half, e to the U is X and then cosine U and the use their necks so cosine of Ln X and finally, we're all done. So I'll just put down a plus C. Right here took carnage in my answer and that's, pretty much it. This right here, it's integral for the original. And now let me show you.
Guys another way to do this, maybe we didn't need the u sub at all and that's check this out right here. And this time we'll actually just do intercooler parts right away, and you'll see what happen. And of course, that's to the TI format, and we'll put on our work right here. So let me put down a D and an, and of course, that's half the plus minus on the side to get ready.
And now we look back this is cosine of our necks. So this is actually just one thing right here right? And of course, I cannot put. This to be integrated because that's exactly what I'm trying to do so what's up being said, I will just put down cosine of our necks right here to be differentiated. And for the icon right here, I'll just put on one and integrating one I will get X differentiate cosine of well.
Next we know that the revert above cosine is negative sine. So be sure you see we have this negative right here because the derivative cosine. And this negative was just the set-up. Okay and continue the input States to say and. The chain rule says, we have to multiply by the derivative of the inside the derivative of Ln X is 1 over X. So just ready done like this.
And now you see I will just do the product of the diagonal that's. The first part of the answer I will write down x times, cosine, X, that's, right at the first right here. And remember when you do the product of a row, that's still an integral so let's focus on this right here. And notice, if you multiply this in that, the X will cancel out, so you actually just have to.
Integrate the function power wise sine of our necks and that's similar to the original one, huh, always know, we have cosine. And now we have to deal with sine maybe works out pretty well. I, don't know, we'll see, well, be careful with the signs here we have minus times minus. So we get plus right here and that's still an integral. And if you multiply this and that you see the X will cancel out. So let me just put down sine of Ln X, DX, right here.
Okay, once again, this X and X will cancel out when you. Multiply them, and you can just put that inside of the integral right here, hmm, how can I deal with this inter, quote, Oh same thing as the original. Let me just do that yet. But this time you see we have to stop right here for DTI method. That was the first run-through right? I.
Am NOT going to continue, because if you continue with the same D column, right here, the roof to this it's going to be pretty bad. So we actually have to stop right here. We focus on this integral. And then we did another DSA that. For this right here.
So here is the D I for that. Integral I'll put on a plus minus on my side to get ready. I am going to differentiate sign of how an X and integrate one similar to the first part, but you had to do that separately you have to do this separately. Okay.
And the reason I knew to stop is that I had a sense of danger. I didn't want to continue. But actually the secret is, in fact, I can deal with this integral by doing this again, that's a deal anyway, enough talking, integrating one, we can. X differentiate is the derivative of sine is cosine.
And the input stays the same and the chain rule says we have to multiply by the derivative of Ln X, which is 1 over X. So let's, put it down like this and that's pretty much it isn't it. And now I will just write this down again.
Here we have x times cosine of our necks. And this is going to be from here, and I'll just do the diagonal and let's see we have these times that. So let me put down the X first and that's, plus sign Ln, the X times X. So that's, X. Sine Y and X, ok, X goes first, and then I will have to multiply by this row and remember that it's still an integral.
So I will actually just write it down in red. Right here. This is still an integral.
This is a minus integral, because you have the minors and the rest is positive. So it's, minus integral. And you see similar situation when we do these times that the X will cancel out. So we just have the inside cosine of Ln, X DX. And you see this right here, it's secretly and repeated situation right? So this.
Is what we are going to do I am going to write this down yet on the left-hand side, so how the integral cosine of Ln X DX. And this is pretty much all that we the other work right here right? And now I will just have to move this to the other side similar to the first method. So I will just add the integral of cosine of Ln X DX on both sides and out to the right here as well. So you guys can comment down below, and let me know which method you guys like better, maybe with the usurpers, or maybe just as. How the SI just straightforward like that anyway, this is a now P cancelled it. And this pasta is two of the integral of cosine of Ln, X, DX and that's equal to this Plus that.
And in the end to carry of this - I'm just going to multiply it by 1/2 right here and I, multiply by 1/2 right here. So finally, we see that, and we're done. So of course, in the end, we can just put on + e and box. The answer and be happy all right.
So this is it, and hopefully gets all like this video. And if you guys do give me a thumb. Up, and please subscribe, thank you so much and that's it.