24 Standard Integrals Mth 102...

Added: 2026-05-05

[00:00:00]Put this position first.

[00:00:17]Are we I'm waiting for us.

[00:00:26]Are we ready?

[00:00:36]Can we see the Can we see the book? Can we see the page?

[00:00:46]Can we see the page?

[00:00:53]Hello guys. If you can hear me, please can you say hi?

[00:00:58]Okay. Okay.

[00:01:00]So, let's wait for orders. Let us wait for orders. Let's just give them like 2 minutes. We start in the next 2 minutes.

[00:01:10]Let's wait for us. I think I have um 12 people now.

[00:01:16]12. I'm expecting more than 100.

[00:01:19]I have 12 people now.

[00:01:27]So while waiting for them.

[00:01:31]So these are the question we dealing with today. We have about 24 questions here.

[00:01:38]I don't know why they are not joining yet. I don't know what they are busy doing.

[00:01:43]I don't know.

[00:01:45]So these are the questions. Trusting God to solve everything.

[00:01:52]I hope you have been hearing what I've been saying.

[00:02:00]Okay, I have 29 people now. 29 people now. 29 people.

[00:02:06]If you can hear me, please just send sticker. Send any any sticker you like.

[00:02:10]I have 33 people now. I have people sharp.

[00:02:16]We start soon. Let's give them one more minute to join. One more minute to join.

[00:02:23]Let's give them one more minute.

[00:02:26]I still have about 33 people here.

[00:02:32]So, we start exactly 4 minutes. This is 2 minutes and 54 seconds now.

[00:02:42]Okay. Somebody is saying if the if the video is blur, please just drop a comment if it's clear. Drop a comment. If it's clear, drop a comment. If it's blood, drop a comment.

[00:02:58]Some people are saying it's not clear.

[00:03:00]Wow.

[00:03:02]Is that so?

[00:03:06]Some people are saying clear.

[00:03:09]Some people are saying clear. Clear.

[00:03:10]Okay. All right. Thank you. Thank you.

[00:03:13]Thank you. All right.

[00:03:16]A guy is still saying his blah.

[00:03:20]A lot of people have said it's clear now. It's clear. If it's block probably your network maybe you should log out and join us again if it's blur can log out and join us again if it's block.

[00:03:34]So okay okay all right all right no problem said it is clear now but it wasn't clear. Now let's start. We don't have time to waste. Can drop your comments, drop stickers, you can like you can react. So make sure you repost you post that link on your status and on your group so that others can see it and they can join. Now this is the first question 1 / x l x dx. Let me write it here.

[00:04:06]Now the first question here tonight is we have a integral of um 1 / x lin x lin lin x dx are together so we asked to integrate this so the first thing to do is that now check we are going to do substitution of variable here I'm going to say Let U be equals to something. Now the convision is now is that how do I know what U is equal to? What our U should be equal to is that U should be equal to a particular stuff here whether is X or Linux or Linux such u the derivative will be able to cancel out another terms. Now if I should say let my u be lin x we don't have time I don't have time to be showing you everything but if I say let my u be lin x I will get something that only cancel out x it will cancel out x alone because the derivative of lin x is 1 /x that 1 /x is going to turn upside down that will cancel out x I still have to do another substitution but to make it much more easy I can say let my u be equals to what l of what lin x Let u be equal to lin of lin x and don't forget now let's find our du / dx. If you want to differentiate lin of lin x I told us then that if you want to differentiate a logarithmic or a lin function just differentiate the inside divided by the inside. Now the inside is lin x. If you differentiate lin x what do you get 1 /x right? Then divided by that inside what is inside lin x. So if you divide by lin x that give you time 1 / lin x right division by lin x means multiply by 1 / lin x. So d / ds give us what 1 / what x l what lin x.

[00:06:12]Is that clear? Is that clear to that part now? Okay. All right. Thank you.

[00:06:18]When we say let u because l of lin x we say du / dx now is what? 1 /x * 1 /x = to what?

[00:06:29]1 /x lin x. Now we are to make dx the subject of the formula because we are changing this function to a function of u. It was in x before. So we want to change it to function of u x the formula. Please if there is anything maybe the video become blur or my voice is cracking just drop it on the comment section. and I will see it then I will know how to adjust. Now let's make dx the side of the formula. Now we say du / dx equals to 1 / x lin x. If you multiply what do you have?

[00:07:07]If you cross multiply you have x lin x du equals to what? dx. Are we together? So our dx now= to x lin x du. Now rewrite the original question back. The original question becomes integral of 1 / x lin x.

[00:07:32]Um you know we already said let our u be lin lin x. So instead of lin x you put what? You put u right? So now what's your dx? Your dx now is equal to what? x lin x du. So put your what? X lin X DU. So you can see now X cancel X.

[00:07:56]Ah the video is becoming blurry.

[00:08:02]Okay.

[00:08:08]I'm coming. Just just wait. I'm coming.

[00:08:15]What of now? What of now? How far? What of now? Is the video okay now?

[00:08:22]Is the video okay now?

[00:08:30]It's still not clear.

[00:08:35]And it's very clear. What's happening?

[00:08:37]What's happening?

[00:08:40]Ah.

[00:08:44]Okay, just just give me 30 seconds. I'm coming. Just 30 seconds. 30 seconds.

[00:09:19]What of now? Is the Is the video clear now?

[00:09:33]Is it clear now?

[00:09:37]Please, I need a response. Is the video all right? Thank you.

[00:09:42]Okay, thank you. Thank you. Now, x lin x lin x are together. Now, what do we have left there? We have integral of 1 * du, we have d u, right? Okay, let's just say 1 / u. We have 1 / u left here. Multiply by what? Multiply by du. Can you see what has happened? Now if you integrate 1 / u, what do you get? You get what?

[00:10:08]Lu, right? That's plus what? Plus c.

[00:10:13]So we've already gotten our answer in the sense that we have lin. Now you initially say let u what of lin x you say let u be l of lin x. Right? So since our final is lin u + c here we are going to get what lin x plus c that's the answer is that first question clear to us that first question is it clear I need our response that first question is it clear to us do we understand the first question that we just solved Hello.

[00:11:01]All right. All right. Thank you. Come come.

[00:11:06]Okay. We are moving to number two. Now I want to I want to use another touch light instead of I think it's clear.

[00:11:22]Now, now let's the second question is we have number two.

[00:11:33]The final answer is lin lin x + c. No 1 / u. If you integrate one / u, you get lin u.

[00:11:46]And since our U list listening yeah since our U is Linux from the beginning if you have a question type it is Linux then since we have linu then it's going to give us lin lin lin x is lin is that clear so that's that for that um please sorry I want to send a link to a group I forgot just give me 30 seconds. 30 seconds. I'm Are we still there?

[00:12:23]Hello guys. Are we still there?

[00:12:28]Hello. I need a response. I told some people to wait for me and have left arrived.

[00:12:37]Hello guys.

[00:12:41]Are we still there?

[00:12:50]Okay. All right. Now I said what if we say let u be= to x and x.

[00:12:56]If you say let u be= to x it will not cancel any. If you say let u= to x it will not cancel anything. The question will just become block like it will become more complex. You can it will become more it will become more complex.

[00:13:15]So you can write it on a note because you don't have time to be testing that and check you will see you can say let you be Linux. You can say let it be Linux but you have to do another substitution.

[00:13:28]But we say next at once what it needs to cancel at at the same time. So that's that for that. Now question two we have integral of with lower limit of 1 upper limit of 27 upper limit of 27 then cube root of x + 2 / cube root of x² this all into bracket square dx are we together I already said he said how do we know which one will be you is anyone you see that if you differentiate is going to cancel out a terms do you get it's going to cancel out terms that will not make the your substitution like the final expression that you want to substitute to be easier for you to solve now for this question now say let u and sometimes in integration you have need to have been otherwise let u be equals cube root of x + 2.

[00:14:40]This one lag. I don't know why.

[00:14:43]Let= to cube root of x + 2. Are we together?

[00:14:49]Hello. Are we together?

[00:14:55]Are we together, guys?

[00:15:03]Okay. All right. Thank you. Now I say let what? Cube root of what? Cube root of x + 2. Now what are we going to do now?

[00:15:14]So what we going to do now? We are going to differentiate. Sorry, let u be = 2.

[00:15:18]We are going to cube root of x + 2. Now let's rewrite it. Let's rewrite it and make it x^ 1 / 3 + 2. Now if you differentiate x^ 1. Now if you differentiate that's du dx = if you x^ 1 you get 1 / 3 x 1 / 3 - 1 you have - 2 over what 2 3 is now what is blow? Oh Jesus.

[00:15:54]Ah, it's very clear here like please if it's clear if it's clear on your side please can you just and some people say it's clear if it's blow on your side please it's network it's network if it's blue on your side log out and log in again so we have one is network at times it's network at times 1 3 x^ - 2 3 right that 1 3 - You have - 3. If you differentiate you get zero, right? So this guy we can write derivative of this guy is same thing as 1 / 3 cube roo<unk> s².

[00:16:41]How did I get that? How did I get that?

[00:16:45]You know if this one you know this is this guy is sending us this not what I should be saying now because we should have been familiar with it. 1 3 um x^ - 1 / 3 * 2 this I'm writing this so if - 1 / 3 goes down it become x^ 1 / 3 which same thing as cube root x and this square is what we have on it you can write this on your note and transform it to this so our du our du dx = 1 / What? 3 cube root what?

[00:17:25]Cube root s². That's our d over dx. Now what are we going to do next? Now um make dx the subject of the formula.

[00:17:37]Make dx the formula. So here what you are going to get if you cross multiply if you cross multiply you have 3 cube root s² du = to what?= to dx. Isn't it?

[00:17:51]Now rewrite the question. The question now becomes integral of what? Our lower limit is one. Why? Our upper limit is what? 27. Isn't it? Now you already say to you. So here you have what? U² divided by what?

[00:18:10]Cube root of what? S. Now what's your dx? What's your dx? Your dx equals to what? 3 cube roo<unk> of 4 s². Now you can see the reason why we are making a particular terms u. Now you can see that this du sorry it has du dx = 3 dx 3 cube root s² du. So this now what happened to it? It cancel this know we have three left and three is a constant. So since three is a constant it can come out. So the question become 3 * integral of what? 1 and with upper limit of what?

[00:18:45]27. What do we have there? U² D U Is it clear to this part before I move on? Is it clear to this part before I move on?

[00:18:55]If it's clear to this part, please can I see your response before I move on?

[00:19:00]Okay. All right. Thank you. Thank you.

[00:19:02]Now what of Okay. All right. Thank you. Now this is what we want to integrate now. Now let me write it at one side here. 3 integral of what? 27 U square what? U² du. If you integrate u square, what do you get? You get u cub over what? Over three, isn't it? Plus c.

[00:19:26]But don't forget there is three outside, isn't it? So if you this one now we cancel this. What do we have left there?

[00:19:34]U plus what? Plus c. So now you you can't leave it like this. Go back to your to your beginning. What's your beginning? that person that said no I I just I saw you just join now and you are saying no so I can't go back because you you are just joining now so you say let u equals to cube root of x + 2 right from the scratch since our final answer is u + c now substitute your u back u cq + c that's cube root of + 2 cub + c this our final answer I hope that's clear this our final answer Now since we get u + c and our u is cube root of s + 2. So we put cube root of s + 2 q + c. So that's number one. That's number one. Now without wasting much of our time let's move to number three.

[00:20:28]Let's move to number three.

[00:20:33]Okay. All right. Thank you. Thank you.

[00:20:35]Now number three. The question says integral of root.

[00:20:43]Oh, sorry, sorry. Thank you. Thanks for that. Thanks for that observation. Put the limit. We have to add the limit.

[00:20:51]Sorry, sorry, sorry. What's our final answer? We have to add the limit this.

[00:20:55]And we don't need to add C again. Thanks you for that. What do we get? We get cube root of x + 2, right? So let's add the limit 1 and 27. Once you definite integral that has limit that has interval, you don't need to put plus c.

[00:21:12]Now let's put it. Now if you put 27 here, we get cube root of what? 27 + 2 minus cube lower limit is 1. 1 + what? + 2.

[00:21:27]What is cube root of 27? What is cube root of 27?

[00:21:32]All cube. All all cube cube root of 27 is 3, right? 3 + 2 that's 5. We have 5 cube minus cube root of 1 is 1. 1 + 2 3 - what? 3 cube. What's 5 cube? That's 125. We have 125 minus what's 3 cube? 3 cube is 9. 125 - 9. That should be 16. That's the final answer.

[00:22:01]Yeah. I hope it's clear now, right? 116.

[00:22:04]We got 116 now. So now let's move to the next question. Let's move to the next question.

[00:22:11]If it's clear, please can you just respond and say it's clear before I write the next question.

[00:22:20]Now please I want us to pay very good attention to this question. Pay a very good attention.

[00:22:29]3 cube is 27. Oh, I'm sorry. 3 cube is 27, not 9. This 5 cube 125 3 cube this is 27.

[00:22:43]125 - 27 that's 98. Our final answer is 98. 98. Sorry. Are we together? Thanks.

[00:22:50]Thanks for that. And that's one of the mistake that you can make during your exam which you should not allow. Please be calm during your exam. I think yeah 98. Thank you. We've gotten it like that. Now the next question. Roo cos x sin cub x dx. Now this question looks this question. Yeah. Yeah. Don't worry.

[00:23:14]It's going to be clear.

[00:23:16]Now this question looks so complex but it's as simple as whatever.

[00:23:25]No, no, no. It's not one way to 125.

[00:23:29]125 minus 27 is 98. It's not 108, it's 98. Now let's solve this. Can we all see the Can we all see the screen very clearly? Can we see the screen before I continue to this? Can we see the question please? Can we see the question?

[00:23:54]Is blow again? What of now? Is it still blow now?

[00:24:00]Yeah, some people are saying yes, probably it's network. But can we see it now? What of now?

[00:24:12]Manageable. Manageable. People say yes.

[00:24:17]Let don't worry. Don't worry. Yeah, some people are saying yes.

[00:24:24]I'm coming. Don't worry. Now it will be clear.

[00:24:28]What of now? Is it clear now?

[00:24:31]Yeah. Thank you. Is it clear now? It's clear. All right. Let's go.

[00:24:37]Okay. Okay. Somebody said it's lagging.

[00:24:41]Not that it's blur. But we can all see the screen now, right? You can see the screen now. Okay. It's clear. It's clear.

[00:24:51]Somehow like more blurry worst. Oh my god.

[00:24:59]I can't see anything. Oh, okay. Calm down. Just calm down.

[00:25:08]Just calm down. It will be clear. Just chill. Chill. Chill. It will be clear.

[00:25:18]What of now?

[00:25:27]Okay. You know what? You know what? Just give me 15 seconds. Just 15 seconds. 15 seconds. I'll be right back. 15 seconds.

[00:25:46]Guys, is it clear now?

[00:25:52]Is it Is it clear?

[00:25:56]Somebody said is clear. It's clear.

[00:26:00]Still waiting for more response. Is clear. Good. Now, great. Thank you. 100.

[00:26:07]Perfect. All right.

[00:26:09]integral of roo<unk> x you can just wait you can just endure at least you are going to learn it even though if time will not permit us to solve everything now how are we going to treat this guy now right this give us I'm coming I don't want to make it blur now this going to give us what integral root cos x * sin cube x can be written as sin x * um I'm coming by um this give us multiply by sin square x yeah sin sin square x dx.

[00:27:11]Do we get that? S I splitted the sin cube x to I was thinking of something before. I splitted this sin cub x * sin square x. So I can write this as integral of roo<unk> cos x mult* sin x are together into what? So the next thing to write here now is sin square s can be written as um I'm coming I'm checking something I hope you are still there I hope you are listening yeah sin square s can be written as 1 - cos² x 1 - cos² x dx is it clear to that aspect now before I move on is it clear to that aspect is it clear to that aspect can we move on.

[00:28:05]I'm waiting for our response.

[00:28:08]Okay. All right. All right. Thank you.

[00:28:11]Now the next thing now is that what are we going to do? What we are going to do now is let's say let's um I'm coming. I'm checking something.

[00:28:22]Okay. Okay.

[00:28:25]I'm checking something. Yeah. Just few minutes now. This this this Okay, let's coming. Something is not coming. Just give me like 10 seconds. Let me think. I need an idea to come now. And if it doesn't come quickly, that means we are coming back to it. I don't want to waste time. I'm coming back to this guy. I don't want to waste time. Now the next question without wasting our time.

[00:28:54]Integral of integral of sec x dx. Now don't worry don't worry I don't want to back that question what I wanted to do is not it's not coming why is this thing blow now somebody is saying blur is it still blur now I hope it's clear ah this thing is please if it's clear please can can I just see your respon if it's clear is it block.

[00:29:34]This one is m.

[00:29:43]Ah, Jesus.

[00:29:47]Ah, and our time is going. Our time is going. Okay, you know what? Let me just let me let me check my setting. Let me check my setting. Just give me 30 seconds, please. Let me check my settings. I think it's Hello guys. Is it clear now?

[00:30:13]Is it clear now?

[00:30:19]Can we hear me? Is it clear now?

[00:30:26]Clear. And we can hear my voice, right?

[00:30:28]Yeah. All right. Thank you. Can we move on?

[00:30:34]Okay. Now, let's solve this. Yeah. For the previous question, I'm coming back.

[00:30:38]I'm going to remember what I want to do.

[00:30:40]Now, sec x. Now, please list.

[00:30:44]Now, there are things that you don't need to question like why why why just get familiar with them? Am I communicating? actually you can question them in the sense that you will need an answer that okay this is the reason why now let me just write what I want to write then I'll now tell you the reason why I'm writing it so what I want to do is that these questions can be written as integral of what se x mult* se x - tan x over se x - tan x dx. Let me confirm if it's going to be minus I'm coming if I integrate this. Okay, I think it's plus I'm going.

[00:31:38]Now what I want you to know is that I've not changed the question have I this over this is one. If I multiply it by this I will get the original question back. Do we get do we get that? I've not changed the question. You still see the reason why I did it that way. You are still going to see the reason. But what I want you to see is that the question has not been changed because what I did is just using one. This divided by this is one. These are the things you should be expecting.

[00:32:09]This over this is one. Are we together?

[00:32:11]So I've not changed anything. Now let's continue. We now see the reason. Just listen. You see the reason. So this give us integral se x * 6 x. What do we get?

[00:32:22]We get se² x se x * tan x. We get what?

[00:32:28]Se x tan x. Isn't it divided by this over one? 1 * this you get what?

[00:32:39]Se x + what? tan x. Then you get dx. Is it clear to that part?

[00:32:47]Is it clear to that part? Before we move on, please I need our respon. Is it clear to that part?

[00:32:55]Hello guys.

[00:32:58]Hello guys. Is it clear to this part?

[00:33:04]Do we understand to this part? All right. Thank you.

[00:33:08]That person that said no just multiply the numerator you will get this. Then multiply the denominator you get this.

[00:33:14]Now I know I gave you guys a law a rule not actually me like I told you about a rule that have an integration if you differentiate the bottom and it gives you the top then your answer is l of the bottom how many of us still remember that like if if um if I'm coming if g prime of x equals to f ofx then integral of f ofx / g ofx dx = to l g of x. Do we remember this? If g if the derivative of the bottom equals to the top, then the integral will give us plus c. Do we get that? So now you can see that if you differentiate se x what do we get se x se x if you differentiate se x is same as se tan x that's why I told initially go and learn all those standard standard derivative and integration if you differentiate x is in your test standard it will give you x tan x if tan x² x dative of tan x give x give you x tan x since the Derivative of the bottom can be find at the numerator. Then our final answer is what? L x + what? tan x plus c. That's the final answer. Without without stressing yourself, you can stress yourself. I spend an hour. But with this with this logic, you are done. Is is the fine answer? Understand?

[00:34:56]I think our response. Do we understand?

[00:35:02]Do we get that? Hello guys.

[00:35:06]Do we get that?

[00:35:11]Okay. All right. Thank you. Thank you.

[00:35:14]Can see it's very simple. It doesn't Now let's I think I remember what I want to do. Now for the previous question.

[00:35:22]No cause for alarm. For the previous question. Yeah, we are correct to this point. We are correct to this part.

[00:35:32]We are correct to this part. Now I hope you are following. Now we say let let u be equals to cos x.

[00:35:43]If you say let u= to cos x then differentiate u du over dx equals to somebody is saying bl.

[00:35:56]Is it blue on our side everyone? Oh Jesus.

[00:36:03]I need I need a response from another person. What of now? Is this still blow?

[00:36:10]This thing should stop blowing now.

[00:36:15]Okay. Okay.

[00:36:25]I'm coming. Don't don't worry. We manage it no matter what. As we are doing it small by small small small doing it small small we finish everything by God grace. No matter how slow we are we finish everything. What is this block?

[00:36:43]Let's check how far how far. Now we go like this. Don't worry. One of the reason for this live class is I can attend to your question live is clear.

[00:36:53]All right. Thank God. Let's move on. Now we say let= to cos x. If you differentiate u, if you cos x, what do you get? Minus sin x. Right? Now make dx of cross multiply.

[00:37:08]du will give you what? Minus sin x dx.

[00:37:13]Isn't it? Now make dx of the formula. Dx will give you what?

[00:37:18]The x is going to give you I'm coming.

[00:37:24]It's going to give you du.

[00:37:28]Oh, wait. Wait. Yeah. If you make the formula divide both side by Yeah. This is going to give you du over what? Minus sin x. Do we get it that? Do we get it?

[00:37:41]Yeah. So now our dx = to d minus sin x.

[00:37:45]Now go back to the original question. Go back to the original question. I'm solving number three. This is the beginning. Somebody saying I should go to previous page. This is the beginning of number for number four. I was before I move to number five. Don't worry. So now let's bring this back to the original question. Now write the question back. Integral of root we already say let u be cos x. Instead of root cos x now we have what? Root u.

[00:38:12]Then multiply by sin x * 1 - cos² x.

[00:38:21]Our cos u is cos x cos square x will be u² dx.

[00:38:29]No sorry we won't put dx again. We will not put dx again. What's our dx? Now our dx is du over over sin x. You know there is minus there we can bring the minus outside.

[00:38:44]Can bring the so this sin x we cancel this sin x.

[00:38:51]So the question now becomes minus integral of what<unk> u open bracket 1us what u² du do we understand to that part?

[00:39:08]Do we get it to that part before I move on?

[00:39:15]Do you understand that part before I move on? Hello guys.

[00:39:21]Okay. All right. Now this can be written root u^ into 1 - what? U² d u minus integral u^ * 1 you have u power right minus you^ half * that give you u d u I hope it's clear see 1 / 2 * 2 that's 1 so u^* u^ 2 that give you so this give us minus If you integrate u^ you get u^ 1 / 2 + 1 that's 3 / 3 / 2 - u^ 1 + 1 / 1 + 1 + c. So this give us minus if you divide something by 3 / 2 is is same as multiplying it by 2 over 3. So we have 2 u 3 / 2 / 3. Are we together?

[00:40:34]This time this you get this. Somebody say I should come again. This times this. Write it in your note. This time this you get this. This time this you get this. Write it and do it. This time this you get this. If you integrate this you add one to it. 1 / 2 + 1 3 / 2. Then that 1 / 2 + 1 will also be at the denominator. We are familiar with this basic rule. this 1 + 1 / 3. If you're dividing something, if I dividing something by 3 over2, same as multiplying that same thing by 2 over 3.

[00:41:08]So I multiply now by 2 over 3. The same thing as dividing by 3 /2 equals to this u 2 dividing by 2. Okay, this give us what? U what minus rather this minus u what u 2 plus c are we together so now what we are going to do next is I think that's all except if you want to find the final answer now is cos 2 2 u what's our u we said initially we said let u be what cos x right so now 2 u cos^ 3 / 2 / 3 - u ² cos² x / 2 + c. That's the answer.

[00:41:57]I added the power. Now, somebody is saying, are we not supposed to add the power? I added the power. Now, I added the power. Let's come again.

[00:42:06]Let's come again. Calm down.

[00:42:10]This is u ra 1 - u². U power * 1 you get U^ U* U² you get U this is it U^ 1 / 2 * U² it's going to give U^ 1 okay okay sorry sorry plus 2 I was using so sorry so sorry so sorry I I could see that yeah yeah so sorry I can see that rather it should be plus I just multiply direct so This is going to give us not u 1 / 2 + 2 that's 5 / 2. Sorry this is 5 / 2. Thanks for that observation. This is 5 / 2. This time this will give us u^ 5 / 2. So this will give us 5 / 2 + 1. Then we have 5 / 2 + 1. 5 / 2 + 1.

[00:43:12]That give us what? Um I think that should be what? 5 / 2 + 1. The SM is 2.

[00:43:19]That give us 7 / 2, right? So this give us 7 / 2. Here we also have what? 7 what? 7 / 2.

[00:43:29]So let's cancel this. Let's cancel this this. So this like simple algebra like what you are just doing are like simple indices. So you can just So we have minus ^ 3 / 2 / 3 / 2 - ^ 7 / 2 / 7 / 2 + c. So this give us - 2 u3 / 2 / 3 - 2 u 7 / 2 / 7.

[00:44:08]Then this give us minus minus. Then we have 2. What's our u cos x 3 / 2 / 3 - 2 cos x 7 / 2 / 7 + c. That's the final answer.

[00:44:32]If you just did, can you just can you drop a comment? If you don't understand, can you also drop a comment?

[00:44:40]If you understand, can you drop a comment? If you don't understand, can you drop a comment?

[00:44:46]Somebody is saying why are we adding plus? We are integrating now.

[00:44:50]We are integrating. That person that saying why are we adding plus one?

[00:44:54]Integration you add plus one to the power. Then we divide by the again.

[00:45:01]Okay. Somebody said he he doesn't understand like two people. Now I would love to because that's that's one of the reason for this live class. It's different from the normal video I post so that you can ask your question. Yeah, I know it can be simplified further.

[00:45:16]Don't worry, we still deal with that. I just want to arrive at it like the first answer first before we simplified. So now let's come back to the original question for those who doesn't understand. Please pay attention. Now cos x then sin cube x dx.

[00:45:31]Are we together? So this give us what?

[00:45:34]Integral of what? Root cos x chera.

[00:45:38]Listen now. Then we have what? Sin x.

[00:45:42]Then we have what? Sin² x dx. Right.

[00:45:46]Which give us integral of what? Root cos x. Are we together?

[00:45:52]Sin x. You know from trig identity sin square x + cos² x = 1. You make sin square x of the formula. Sin square x you give what? 1 - what cos square x so this sin square x here we can change it to 1 - cos² x I'm doing it again that say so now this is the this is what we've obtained now are we together so say let u let u be equals to what cos x so now if u= cos x then du / dx d over dx will give us whatus - what?

[00:46:32]Sin x. Make dx of the formula. If you cross multiply, then du will give us what? - sin x dx. Then that means dx will give what? du over what? Minus d over what? Sin x. This one. Now don't tell me this is just like your normal primary school division. If this is this make one of the formula, you get this.

[00:46:56]Now bring it back to the original question. The original question becomes integral of root. Since we have already selected you because X still blow now. Is it still blow now?

[00:47:13]What of now? Is it still blue?

[00:47:17]Is it still blow now?

[00:47:20]I'm I'm waiting for a response. What of now? Is it still blow?

[00:47:26]Still the same. Okay, don't worry.

[00:47:29]Don't worry, we do tonight class. Don't worry, Alpha. Is it clear now? Blah blah blah.

[00:47:51]Sheep, is it?

[00:47:58]Is it clear now?

[00:48:01]Yeah. Sharp. Somebody said vanishing point. Okay. Now the question is root cos x. But since our cos x is already, so we have root u multiply by what? Sin x.

[00:48:15]This 1 - cos² x cos x is u. This becomes 1 - what? U². What's our dx?

[00:48:24]What's our dx?

[00:48:27]It's not about Tik Tok or YouTube. It's my phone. I manage my phone like this.

[00:48:31]No vest. It's not Tik Tok or YouTube.

[00:48:34]YouTube is the best. Not my phone. My phone no good. But let's just run. So since our dx is minus du instead of dx let's write minus that minus we can bring it out. So what do we have? What? Sin x. So sin x cancel sin x.

[00:48:56]Clear. So the question now becomes minus integral of what roo<unk> u * 1us u² which will eventually give us integral of =us integral of u power.

[00:49:20]Yeah. Yeah. The minus is constant. It's just like minus one into 1us. It's not it's not a variable there is constant du. So this give us what - u power * 1 u power minus u^ * u^ 2 just like say u^ 1 / 2 * u² it's going to give you u^ you know ah Jesus hey lord have mercy oh Hey, Alpha.

[00:50:17]What the are we together?

[00:50:25]It's clear now. Now u^ * 1 we have u u^ * u² this is it at the top u^ * u² so that g^ 2 right which is ^ 5 / 2/ + 2 is 5 / 2 find the SCM do it are we together okay okay all right so this give us what^ 5 / 2.

[00:50:59]Do you? Now, now we want to integrate.

[00:51:03]Yes. So, we have minus integral. If you integrate this, you get half + 1. Half + 1 that's 3 / 2. Then you divide by that + 1 againus u power 5 / 2 + 1 that give you 7 / 2 again.

[00:51:25]Sorry. Sorry, we are not adding integral again. Sorry, it's bracket now because you already plus what? Plus c. Is it clear to that part? Now it's not u. It's u square that is there is not u. U* u².

[00:51:47]This one is one. This one is u square.

[00:51:51]So now so the question now becomes minus open bracket u^ 3 / 2. Now dividing by 3 /2 is same as multiplying by 2 / 3us ^ 7 / 2. Dividing by 7 /2 is same as multiplying by 2 / 7 plus c. Isn't it? Now 1 / 2 is common to both. 1 /2 is common to this one. So we can change it to root and two is also common to them. Minus rather two now this guy first they see blood is it blow now again.

[00:52:34]Ah Jesus.

[00:52:44]What of now? Is it blow?

[00:52:58]We go rough. We go rougher.

[00:53:02]Meh.

[00:53:04]The next one I will do I will try and get like laptop and instead of all this one. So now here we have two.

[00:53:14]So this becomes u power 3 - u 7 open bracket 1 / 2 this over 3 this over 7. I'm going to explain plus C.

[00:53:37]Can we see what I did here? Two is common to both of them. Two is common to both of them. Bring the two out.

[00:53:46]And power of half is common to both of them. Extract half out. See 3 * half you get over two back. 7 * half you get your 7 / two back. Do we get? So extract the half out. This one will remain seven.

[00:54:02]This one will remain three. Also bring this two out. You bring it out here. See that?

[00:54:10]Now this now becomes this now become um minus Chinese language.

[00:54:22]Are you joking or somebody saying is showing Chinese language?

[00:54:27]We move through the bloods and declare So here we have open bracket.

[00:54:36]Let's bring the two out since two is also a constant. So we have minus 2 open bracket root.

[00:54:45]Why are we having root? Because there is a power of half here. So you can now have root. Root u cub - u7 / what? 7 + C then you can now put your this now becomes what - 2 open bracket root don't forget your C plus C is outside are we together so here what do we have we have what's our u cos x right cos x.

[00:55:27]So this becomes cos x / 3 - cos 7 x over what 7. This is the final answer. And you can decide to factor out cos x. You can change it to cos x into minus cos 4x whatever. Do we get and you can find your LCM here. Did we get Is it Is it clear now?

[00:55:57]Is it clear?

[00:56:03]No, it's minus. It's minus. Not not plus. It's minus.

[00:56:08]It's minus.

[00:56:10]The next question.

[00:56:16]We move to the next question.

[00:56:18]Sharp. Can we move to the next question?

[00:56:23]Hello guys. Can we move to the next question?

[00:56:30]Okay. All right.

[00:56:32]Now the next question. I hope there won't be blood ship this time.

[00:56:39]The next question on the list now says, "Now we want to perform another miracle.

[00:56:45]Now let me use the touch.

[00:56:49]I'm coming. I want I want to start the next question very clear.

[00:57:03]Can we see clearly now? Can we see it clearly? Can we move on?

[00:57:10]Okay. Now the next question. Integral of integral of now please pay attention 1 / 1 + sin x dx 1 + sin x dx. Now we want to perform a miracle. So this becomes integral of over + sin x.

[00:57:42]Now what we want to do now is just like rationalizing you know your salt then so it's just like want to rationalize but actually I don't think that's called rational cuz this is trick now so here we are 1 - sin x over 1 - sin x have I changed anything yeah I want an answer I've not changed anything cuz this is just one. So it's just like the original question back. Do we understand? Is that clear? Can we move on?

[00:58:21]Hello.

[00:58:23]Let's move on.

[00:58:30]Just the I mean thing say that two that is common to the power I carried.

[00:58:39]Okay, it's clear you know about Yeah.

[00:58:41]Yeah. Yeah. Somebody can say how will I know? How will I know? Now another issue is that sometime is based on the fact that you are familiar with what to do.

[00:58:51]This is not the only way of solving this. We have another method. But this is the simp one of the simplest method of solving it. Now whenever you have s whenever you have s right whenever you have s what do you do? You rationalize to make it by multiplying both numeric rationalizing. Now somebody said only God save us from this blood ship.

[00:59:19]Now yeah now yeah I love that. I love that idea. Somebody say why don't you say let you be 1 + sin x. Now if you say let you be 1 + sin x. Now differentiate it you will get you will get what cos x then what you now be having is that you now be having x function and u function inside single question you'll be having x function and u function inside single question because nothing will cancel out what you differentiate there's nothing else there because this one we already done to you when you differentiate x nothing will cancel out that cos x you now be having cos x / u dx or whatever and that shouldn't be do we get that so now if you now multiply it like this now if you see this kind next tomorrow you will know what to do that's one of the reason why so 1 * this you get 1 - what sin x / this like difference of two square right difference of two square one 1 * - sin x will cancel if you use sin x over sin x you have issue. This is the easiest way. If you use 1 * - sin x will cancel sin x * 1.

[01:00:37]Then sin x * - x will give you minus sin square x.

[01:00:44]We see you see it. I'm coming dx. This guy always say blah blah again.

[01:00:55]Is it clear now? Please do it.

[01:01:00]Check what I did. Check what I did. Is it clear to this part?

[01:01:06]Do we understand to this part?

[01:01:09]Do we get to this part, please?

[01:01:13]Can we see what I did? Yeah. Thank you.

[01:01:16]Just multiply this. Write this in your notes and multiply. Get this. You will get this. This is the advanced level for crying out loud. Now, now this can becomes integral of 1 - sin x over what is 1 - sin square x?

[01:01:36]Don't forget that's dx, isn't it? Now use trig identity. 1 - sin square x is cos square x. Can we see? Now you can now split this integral into what? integral of 1 / cos² x - sin x / cos² x dx.

[01:02:04]Do we get that? Is it clear to this part as well? I think that should be clear.

[01:02:10]Now let's let's split further. We have integral of 1 / cos² x - sin x / cos x * 1 / cos x.

[01:02:27]I think it's also clear to this part now. Is it also clear to this part?

[01:02:33]Is it also clear to this part?

[01:02:34]Somebody's asking me why am I splitting?

[01:02:36]Why will I not split? If I don't split, I do.

[01:02:41]Check. Check that person that say you don't understand if I don't split what can I do? Do we get to this part? If you have this normal mathematics now if you have what is blur again before we do anything somebody will be saying is blow why you have an issue with this thing why is this thing is this thing a problem 100 level student from year to year how is it a problem if you have a minus b / c is same as a / c b / And that's what I did here. 1 - sin cos square x give us 1 cos square x - sin cos square x. So how that should not be a question tonight. Do we understand?

[01:03:33]Do we understand what I did?

[01:03:36]If you substitute directly you will have issue please. But is it clear to this part?

[01:03:45]I'm asking. All right let's continue.

[01:03:48]Now this give us what integral what is 1 / cos² x that's se² xus what is sin x / cos x that's tan x* what is 1 / cos x that's se x dx.

[01:04:06]So this give us integral of se square xus se x tan x dx Now let's integrate. Now you will see like all those people that are saying why don't you substitute try try it. No I've seen because we don't have time. I should have testing other method.

[01:04:30]Yes. Thank you. Thank you guy. Thank you for that that guy that said we splitted because it's forming s square x and cs tan x. These are simple intrigue. I love that com. Kudos. So now if you should integrate square x you know if you differentiate tan x you get s square x if you integrate if you should integrate s² x what do you get tan xus if you inte x if you integrate se x tan x if you integrate se x tan x what do you get se x different x you get se x tan x tan x you get square x so if you integate them you get Please plus C. That's the final answer please.

[01:05:16]Is it clear now? Is it clear?

[01:05:24]Is it clear? Before I move to the next question, that person is saying no. Point out where it's not clear to you. Point it out where it's not clear. point out.

[01:05:44]Okay. Clear. Thank you. Thank you. That guy said no. State your no. So how do we know when to use sub subscription and this? It's not about the thing is check the structure of the question you know and give an assignment. Now write this question in a note. Solve it to this point and think of what to substitute.

[01:06:05]You are raising finger. Drop your question. If I say let because x and what will I get? I get minus cancel this.

[01:06:19]Why? Because it is cancel.

[01:06:24]I think of how to substitute it. You see that something.

[01:06:30]But you see something like this.

[01:06:33]You know there is a way there is something you can multiply with this to give you something like difference of two square and this guy has an identity of cos square x split it split it then you can still remove one cos x from here sin x cos x then 1 / cos x then we're done.

[01:06:55]Okay, let me also do my thing so that it will be clear now.

[01:07:06]>> I hope we'll be able to Okay. All right. Yeah. Yeah, it's clear now. It's clear now. I believe it's clear now. Can we move to the next question?

[01:07:21]As you keep moving for the next question having more problem because the next question is like more complex than the previous one. Now the next question is we have integral of root a - x / a + x dx. I hope we can see this very well. So what do we do now? Let me split this thing into two.

[01:07:54]By God's grace. Yeah. Yeah. By God grace.

[01:07:59]Now here now I want to rationalize. I want to rationalize like I want to rationalize actually the just see what I want to do.

[01:08:10]This one give me integral of roo<unk> a - x / a + x multiply by this is the question somebody saying I should write it this is the question here multiply by now I want to use the denominator to rationalize so I'm going to use the which is a - X root root that part this divided by. Do you understand what I did?

[01:09:02]Do you understand what I did here?

[01:09:06]Check the question. I've not changed anything just if you can just question look you will see what's what's happening I've not changed anything I'm just rationalizing that's just it are we together so now the next thing to do now is now write again integral of now you know if you have root a b is same as root a what root b so this numerator now is just root a minus x * roo<unk> a - x okay now If you have roo<unk> a * roo<unk> a, what gives you? It gives you a, isn't it? So roo<unk> a - x *<unk> a - x, we have a - x. Am I lying? No, isn't it again? I'm coming. Let me do my thing through the blood and the clear is your is your network is your network.

[01:10:07]Don't disturb us again. Root a minus s * root a minus s you get a minus x right.

[01:10:13]So roo<unk> a + x * roo<unk> a - x we're going to get there.

[01:10:22]So it's going to give us root root a + x a - x dx. So this give us integral of what a - x / root if I should open get a² - s² dx. Before I proceed, is it clear to that part? Is it clear to that part?

[01:10:54]Before I proceed, is it clear to that part?

[01:11:00]I need our respond bracket. You know what is paining me?

[01:11:16]Some people are having issue with something that they should have known.

[01:11:20]I'm not I've not started integration.

[01:11:22]That should be the problem. Rational together. This is 100 level.

[01:11:30]100 level. I told us go and learn some things. Go and learn some things. Go and learn this. Go and learn this. This you get this. You get this. Write it in your note. Do it separately.

[01:11:42]Thanks for that. Thanks for that guys.

[01:11:46]two square. Now, can we proceed?

[01:11:50]Now, um there is something you need to know.

[01:11:54]Now, I'm coming. I'm trying to check.

[01:12:01]Is it going to work out?

[01:12:04]Wait, wait. We are not splitting yet.

[01:12:05]Okay. Yes. Thank you. We are splitting.

[01:12:07]Thanks for that. We are splitting.

[01:12:10]So, this give us integral of Yeah, we are splitting. We are splitting. A - and sorry integral of A over what? R A² - S square BL. Is it blur? Please others should reply. Is it blur? Now before I do my thing again, is it blur?

[01:12:32]I'm waiting for a reply.

[01:12:36]Sh. Oh yeah, I'm coming. I'm coming.

[01:12:40]I'm coming. I will do my thing. I will do my thing.

[01:12:52]and clear it tonight. What now? Is it blow now or is clear up? What's happening? Is it blow clear?

[01:13:02]All of now. Is it blow? Clear.

[01:13:09]It's clear. All right. Good.

[01:13:12]Now let's move on.

[01:13:15]So now let's split. Let's split. So this becomes - x / what? Root a² - s² dx. So let's split each. Let's split them to each integral. So this give us integral of a over what? roo<unk> a² - s² dx - integral of x / what<unk> square - s² dx before I move on is it clear to this part this a² sorry yeah is it clear to this part before we move on is it clear before we move on is it clear to this part I'm waiting for us okay all right let's move.

[01:14:06]Okay. Now the next thing now. All right.

[01:14:08]Thank you. Now please take note. Now pay a very good attention. Pay very good attention. There are some three rules like rules that I want to give you. Now these are the rules. Let me write it here. I'm coming. I'm bringing another sheet paper so that I can write it.

[01:14:26]Please can we see this sheet of paper very well? I hope it's not blur.

[01:14:30]I hope this sheet of paper is not blown.

[01:14:32]Now, is this sheet of paper blur? This new one that I just Is it blur?

[01:14:42]Is the paper clear? Let's talk now.

[01:14:46]Okay, it's not blue. All right. All right. Now, now if we have root a² - s² use use x a sin a sin theta cr them but I will still show us how we can know them without cring. I will show us how we can know without cramming. If you have roots a² + s² use x = to a a tan theta.

[01:15:28]If you have root s I'm still going to confirm whether I'm right s² + a² use x = you're going to use s = to um a se theta now these are the three things you need to know yeah trick substitution yes these are the three things you need to know always tell us you have to know them know them often but I try to ask question can we just No, in case you forget, can we derive them? So, how do you derive this? How do you derive this? I don't Please, is it clear now? Because I don't want it to become blur once I'm dering this. I hope everything is clear now. Is it clear?

[01:16:18]The difference is this one is minus, this one is plus, this one is x before a, this one are a before x. A before x.

[01:16:26]Are we together? Now let's move on. Now we are using the the trig identity cos² x + sin² x = 1.

[01:16:38]What's the second one? If you divide everything by cos square x. The second identity is 1 1 + tan tan² x = = 6² x. Now the third one if you divide by sin square you get c square x square x I'm coming square x + 1 = 1 / x cosec cosec x Yeah these are the three standard identity before I still move I have not started the proving these are three identity before I move on to show the proof how we come about this I want to ask again I hope the video is clear is it clear now can we see the board can we all see before I move on please I'm asking us question now okay all right thank you now let's move Now here here you know that here you have cos² x = 1 - what sin² x cosec square x I I don't even think I need this one but this is an identity that you are familiar with don't ask me this this three identity no no it's not a question sin square x so now you can see at listening listen this thing looks like a here is a constant s² we can write this 1 - sin square x as 1 square - sin square x are we together so here you can replace this in form of a² - s² like it's just a logic It's just a logic do we get? Cuz a here is a constant. It suits into this like we can make cos square x to cos a square - x² in since cos square 1 square - sin square x in this. So in a sense in a sense this we want to since this one are equal 1 square= to a square in a sense not that they actually equal. So you want to make these two also to be equal. Are we together? So if you want to say sin square x is s² that means sin x all into bracket square = x² isn't it?

[01:19:41]So yeah we conclude that sin x = x. What is blue again? What's What about blist?

[01:19:57]So here you can conclude that this equals to this since here we have one. So let's say this the coefficient here is one. So here our x = to sin x since there's a coicient of one. This is one. This also one. But here instead of one we having a. So we say let a be x be that's just the logic maybe if I have if I have okay thanks you thanks thanks for that guy it act as a variable now let me show us the second one so that we can use it to get the idea of the first one now we have a square + s² now go and check the identity again you have 1 + tan square x equals to something we are not concerned with what they equal But we are concerned that this thing is like 1 + x is a valid identity that is equal to something. The s square x is equal to we are not concerned with the square x but 1 + tan square is a valid identity. So and we can replace it with what 1 square + tan square x. So since this question is a square + s². So we want to use a square + s² = 1 square + tan square x.

[01:21:20]So yeah you can say these are constants.

[01:21:23]So you can say this equal to this in this can say x= what tan x are we together? Do we get that? So but because here instead of one we have so we can say a tan x that's just like the idea.

[01:21:37]Do we get so if you cannot me I don't know how to cram this thing. If I forget, I'll just use that this this trigonometry just remember back. So, and I think it makes sense that way. So, but nonetheless, you can just try and just spam it. If you have constant minus s², use s= a sin theta constant s², use s= a tan theta, s² + constant, then use s= a sec theta. Just know the move on.

[01:22:10]Now is it clear now? Is this clear before I move on?

[01:22:16]Can we see the board very clear now?

[01:22:20]Can we see the board very clear please?

[01:22:23]Hello guys.

[01:22:27]Hello. I need our response.

[01:22:31]Clear. All right. Let's move now. A is a constant. We can bring it out. Let's leave this for now. Let's focus on this first. We are still coming back to the second one. Let's focus on this. This becomes 1 / roo<unk> a - s² dx. So let x = what? A sin what? a sin theta.

[01:22:57]So now substitute a sin theta for x. So this one becomes a integral of 1 /<unk> a²us our x now is a sin theta² dx to a integral of 1 / a² - a² sin² theta oh sorry this dx we need to change it to d theta. Now since you've changed all the variables from x to theta, this ds also must be changed to d theta. I'm coming back to that. Let me write this first. Let's just write the dx. I'm coming back. So let me write it here. We have a integral 1 / what is common to these two. A square you have a square sin square theta. Now let me not write the s first. Let me not write the s first. Now we can see that our x= a sin theta right now dx d theta. Now if you differentiate if x= to a sin theta if you differentiate the x / d theta will give us what will give us a theta isn't it? Ifate a sin theta what do you get? A cos theta. Now make the x of the formula. The x will give you what? A cos theta d theta. Are we together? So instead of writing dx here, I'm going to write a cos theta d theta.

[01:24:35]So this a2 can now come out because it's also a constant. Are we together? So here outside I now have a square integral of 1 / a².

[01:24:47]Yeah, sorry there's a square root there.

[01:24:50]Sorry about that. Square root is there.

[01:24:52]Thank you. 1 / a² into bracket 1us. Now 1 - sin square theta is what?

[01:25:04]cos² theta. Isn't it? Here we have cos theta d theta.

[01:25:14]I think it's making sense now. So here we have a square integral of now this is root a square this is so okay let me split it cos theta / roo<unk> a² * roo<unk> cos² theta roo<unk> cos square theta d theta root a square is same as a root cos square theta is same as cos theta are we together let me look for because I don't want to move that let me look for the small paper to finish that one.

[01:25:49]So yeah, what do we have? So here we have a cos theta over what's root a square a.

[01:25:59]What's root cos square theta? We have cos theta isn't it? So yeah a cancel one of this d theta cos theta cancel this.

[01:26:10]What do I have? A integral of what? 1 d theta.

[01:26:15]If you differentiate one with respect to the what do you get? A multiply by this blah blah blah.

[01:26:35]Is it clear now? I will come again. Is the video clear now? Is the video clear?

[01:26:40]Please can we all see it now? Please.

[01:26:50]Thank you. Now listen.

[01:26:54]So yeah, what do we have? We we have what as this video no clear just they pay me.

[01:27:01]Okay. You know what? I don't like all this blah blah blah blah blah stuff.

[01:27:07]Um, should we because of this something and blood there and blood that should we should I record the video because I actually want us to interact like you can ask question then I can answer you.

[01:27:25]Should I record? Should I record the video and upload tomorrow and stop this one? This one that is blowing or we should continue. What do you think?

[01:27:38]Yeah, I'm waiting for a response. What should we do? Should we continue?

[01:27:44]We should continue. Type Don't type yes or no. Don't type yes or no. I wouldn't know because I I've asked two different question.

[01:28:01]Okay. Majority carry the vote. Now, let me create a poll. Let me create a poll.

[01:28:05]Don't worry, I will create a pool now.

[01:28:07]Where is that pool stuff?

[01:28:11]I'm trying to create a pool before the next live class. We are going to have before the next one. I would have I would have um what was it called?

[01:28:33]I would have gotten a I'm coming a good phone or a laptop. I want to create a pool. I'm looking for a poll. Majority carry the vote. Don't go yet. Just don't go yet. Just wait. Where is this stuff? Is this thing?

[01:28:56]I only see this pool again.

[01:29:12]>> Just wait. Wait. Don't go yet. I'm looking for you guys. Just queue. I'm coming.

[01:29:30]Some people are just saying upload s up.

[01:29:33]Some people are saying we should continue.

[01:29:37]And another issue I have with uploading is that if I solve this question and I want to upload it, it cannot take me anything less than 20 gig cuz I will use nothing less than 2 hours. the p fraction I uploaded this morning I use 8 gig and it's just 54 minutes. So that's another issue I have with uploading.

[01:29:57]I've just been trying to just to just try and just do something so that you guys can have something to watch. It's not actually easy. I'm looking for the poll. I can't see it. It's It's kind of see um just give me I need to create this poll.

[01:30:28]I've been looking for it.

[01:30:31]Yeah, I've seen it. I've seen it now.

[01:30:33]I've seen the P.

[01:30:36]Yeah.

[01:30:39]Should we continue or I should record and upload continuous we we Manage the blow.

[01:31:12]Manage the manage the blush.

[01:31:56]Let's vote.

[01:31:58]Hey, let's vote. Let's vote. I'm waiting for our voting.

[01:32:07]Let's vote. Let's vote.

[01:32:10]Let's vote.

[01:32:12]Let's vote.

[01:32:16]Let's vote.

[01:32:18]Yeah, we are fighting. Let's vote. Let's vote.

[01:32:22]Let's vote.

[01:32:30]Can we hear my voice? Please, somebody should drop a comment. Can we hear my voice?

[01:32:36]Can you hear my voice?

[01:32:39]Yeah. The campaigning. Can you hear my voice?

[01:32:44]Now, one thing is that the video, this video, what we are doing now is actually recorded. It's recorded.

[01:32:52]It will be uploaded automatically on YouTube. You will have access to it. But the issue I'm having is we spent close to 2 hours yet. I've not solved.

[01:33:03]Yes, I'm to we spent up to two hours yet I've not solved more than six question out of 24. So imagine we saw six question out of 24 and we spend like 9 minutes six how many six can we see in 24 that's four times 90 minute time 4 we can see that's 360 minutes that's about 6 hours that's about 6 hours so do we get that so we spend 6 hours and yet we still be passing through blur and blow and clear and blur and clear and blow of course but me I feel like we should either continue and also feel like maybe we should just pause then I just do the normal uploading actually the reason for this live cry so that okay I can see our inter then yeah let's vote now the vote will end in the next 1 minute majority carry the vote the vote will end in the next one minute but The thing is that the next live stream the next live stream I'm going to make sure whatever it takes I'm going to make sure I'm going to use laptop like this my phone don't try going don't suffer for you the phone don't try so the the I'm going to make sure we are going to use laptop so the vote is stop in the next 1 second 30 It goes.

[01:34:47]20 seconds left.

[01:34:51]Continue. You are losing.

[01:34:54]Those those who are voting, continue.

[01:34:56]You are losing. They winning. You 15 seconds left.

[01:35:04]Now the thing is that to avoid to avoid you trust me now she will trust me now I record it you will understand like clear clear don't that one is not a problem I will do it you guys will understand perfectly like very perfect that's not a problem but the reason for this live stream is so that we can you know we can get it very well and so I want to create another poll I want to create Another pool.

[01:35:38]I want to create another pool.

[01:36:13]I hear the Okay.

[01:37:05]Okay. Okay.

[01:37:12]Oh yeah, I'm just laughing here.

[01:37:17]Actually I love I love live stream because the interaction is kind of lively like in fact at times the way you people fight in the comment section like that guy do ah boss man ah thanks for all those kind of interaction it's kind of do we get it's kind of in fact I feel like even though without continue I feel like we should just continue chatting yeah and just continue talking but nevertheless now like question six out of four. So we kind of stop here but I promise you before you wake up tomorrow morning you will see the video on YouTube. I would have made everything ready. The 24 questions solved clear and clear.

[01:38:07]Yeah that's it. Live stream gives room and that's why please please you know that live stream actually give room for questions. Yes. of which I already said in the next live stream I'm going to do I promise you can anticipate I'm going to make sure I use laptop so that there will not be blah blah there will not be do we get do we understand that to do it that's one and two whenever you watch my YouTube video and it has helped you like it don't just watch and leave drop a comment section and drop a comment to the comment section do we get drop a comment drop a comment whether you enjoy enjoy it or you don't enjoy abuse me in the com you you did rubbish you record rubbish can see anything so we know how to just just always drop your mind in the comment section do we get so um no no no no no recording shop data past live stream that guy that say live stream the shop data now for this live stream we've been doing since probably I don't think I would have spent more than 700 MB on one gig yeah if I should solve everything tonight on on this live stream. Even though I'm going to use 6 hours, I can't spend more than 3 gig maximum in streaming this.

[01:39:20]But if I should record it normal video, 6 hours 6 hour is going to like 70 gig. I'm not exaggerating.

[01:39:29]But I can I can manage 2 hours to solve everything. Normal video can manage 2 hours. That 2 hours now, trust me, it's close to 20 gig. I'm not lying. You'll see tomorrow. I will post it when I'm uploading. You see that it's very close to 20 gig. You will see it. So live stream is another thing I prefer live stream that it saves me of data. Do we get and once we are done live stream YouTube will just upload it there without using my data again I just need to data like they won't use the we get all that so live stream is actually better in terms of data consumption can leave now be expecting the video tomorrow early morning tomorrow early morning very early in the morning if I can wake up to watch it before you go to your class are we together So thanks.

[01:40:18]See you. You can leave. You can leave.

[01:40:20]You can leave.