I Tried ISI Integration Bee 2026 Top 8 Bracket Round

Added: 2026-05-28

[00:00:00]What'sabi guys? Silver here. We are going to try ISI integration B round of eight. I think it still considers like still regular round. Yeah, I think we're still kind of in regular round technically. Let's see what we got. Are y'all ready for the round of eight integrals?

[00:00:21]Let's start.

[00:00:24]Oh Okay.

[00:00:29]So, I'll I'll do I'll do two ways for this. I'll do one for the advanced and one for the intermediates. The one for advanc is very easy, right? If you remember your uh your advanced integration v training, right? We can have squared loachvky, right? We can absolutely have squared lobachevsky. And this is what we have, right? So what this means is that this is p<unk> / 2 sin square and just like even honestly even like you can even solve this mentally like you can mentally say that this is pi over4 uh that's for the advanced so square loaches very OP uh for the intermediates uh my condolences because you have to do a bit more work unfortunately it's it's unfortunately more work for the intermediates you have to format this in ditch integrals. So you have no other choice but to bash. So oops I'm sorry. So whenever you see something like this I would always recommend integration by parts until you have like 1 /x.

[00:01:40]Okay. And then of course this is like 4 sin cub x cosine of x. And so what we have now uh into infinity is going to equal to zero from zero is also going to equal to zero. So what we pretty much have is just from 0 to infinity all of x sin cube of x cosine of x.

[00:02:04]Okay, this is where the things is just kind of annoying. I'm sorry intermediates, but we're going to have to just kind of bash some derive integrals here. And what I mean by that is we have to utilize trig identities to kind of help us out. So we have sin square and we have like sin 2x. We'll take we'll borrow the two from this.

[00:02:30]And then we'll go ahead and turn we'll go ahead and turn uh 2 sin square or let me put the two there so it's less distracting.

[00:02:40]We have sin 2x and sin square can be turned into what?

[00:02:46]Half minus half cossine 2x right the general senses is just to get everything into in forms of uh ditch integrals as much as we can of course.

[00:03:00]So okay so this two kind of casts out with these. So, we are left off with like sin 2x minus um a half of sin 4x. I'm going to go like this.

[00:03:19]Okay. Maybe it's not as bashy as I thought. Okay. Well, well, that's good. Thank goodness. So, that means like what? Thankfully, thankfully, it doesn't matter what you have inside the sign, right? Let a be literally any number. This will still equal to pi /2.

[00:03:40]And the reason why this works is because of this x. That x cancels out the the 1 / a thing. When you let u equal a of x, right? This becomes like u over a. And that cancels out the a, right? So it's just going to give you the same der. So it doesn't matter what you have inside.

[00:04:02]It's going to equal to pi / 2. Okay. So with that in mind, we have what?

[00:04:12]Oops. We have pi / 2 minus that's a half of pi / 2 pi / 4.

[00:04:23]Okay. So that's the intermediate level way of solving this is by spamming uh or bashing with ditch integrals. Okay. So whenever you see x square at the bottom uh integration by parts to turn this into 1 /x. Okay. So yep. But square loaches is very OP for advanced users.

[00:04:44]You can just use square. You can mentally use squared loaches. And you can already see that this is equal to pi over4.

[00:04:52]Yeah. Integration by parts. This is the This is the um the intermediate. Oh no.

[00:04:58]What did you do?

[00:05:01]Oh no.

[00:05:03]Yeah. No, no, no. Do do the do my way.

[00:05:07]Uh but yeah, this is Yep. Per four. You got it. Squareepsky is very OP. Very OP.

[00:05:16]All right. Awesome. That bashing I thought that turned out a lot nicer than I thought. Honestly, I'm really glad for the the derit bashing was it turned out nicer than I thought. I thought there was going to be more, but uh but most most commonly it's it's more uglier than that. But thankfully in this problem, this integral was a lot nicer. Okay, next one. What is this?

[00:05:41]What is this? What is this? Let's let's play around with this. So let's let's factor out the x squar. It's like x4 cuz what is what's going on here? x5 - 1^ 2.

[00:05:55]So many x to the powers of fives.

[00:05:59]Oh, okay. Wait, wait. But we have x5 x 5 - 2 x5 - 1. What happens if I let u= x ^ of 5 - 1? Thankfully, that's an odd power, so we're not going to nothing bad's going to happen for us. uh this five. So this is going to be 10 over 5 uh u^2 e to the u u minus 2.

[00:06:29]Wait what?

[00:06:32]E root pi letting u= x ^ 5 - 1 gives immediately gives me this u ^2 of e to the u u - 2. Oh, wait. What am I doing? What am I doing? What am I doing? What am I doing?

[00:06:52]Oh, what am I doing? What am I doing?

[00:06:56]What am I doing? Oh my god. U^2 E to the this is X^ 5 as X^ 5 - 1. So this is U + 1 and then this is U minus one.

[00:07:15]Oh my god.

[00:07:18]So now this is u ^2 + 1. Oh my goodness.

[00:07:25]So this is just 8 root pi.

[00:07:30]There we go. That sounds about right.

[00:07:34]Jeez, I am rushing too much. Please be careful with that. Okay, just please be very very careful with that. Second, when I first looked at it, I thought it was going to be like integration by parts. I don't know if integration by parts would work or probably would. Just takes longer. Next integral.

[00:07:56]Oh, what is this, bro? I don't want to do this.

[00:08:03]That's disgusting.

[00:08:05]I don't want to do What the absolute value. Yeah, you're funny. I don't want to do this. Let's Let's take a look at the solutions, shall we? Cuz I don't want to do this. It seems like you might think this is equal to like what? Like if you do let u equal inverse tangent of x then you might think oh well this is just gonna equal to 3x right wrong you're wrong it's not going to equal to 3x because of the domain you got to be careful you got to be careful with these okay I'm pretty sure this surpasses one surpasses two obviously so you have to be very careful because it surpasses pi / 2 you have to consider like some sort of dom main restriction or something and I don't want to deal with that. I don't want to deal with that. That's disgusting.

[00:08:58]How do you guys do it?

[00:09:01]Ely split it in three different parts.

[00:09:03]You see, it's nasty. You do not I I'm not interested. Uh I'm too lazy.

[00:09:12]I don't want to do graphing. But yeah, so you split it in three different parts. And then with this a branch cut.

[00:09:20]Ew.

[00:09:23]Yeah. And then you have to split it in certain intervals because the absolute value and uh gh disgusting. It's a very nice answer. Very nice answer. But it's it's brutal. It's a brutal graphing integral.

[00:09:42]Yeah. I'm not going to be able to do that. Okay. Let's Let's move on to the next integral.

[00:09:49]Oh god, what is this?

[00:09:53]Oh jeez.

[00:09:56]Well, more peace-wise integrals. I'm not surprised, but this one is kind of uh sus. I would say it's hard to separate or to sum it. I might have to do this constructively from one to let's let's do it one by one. Let's let's do each one by one. Right? This is easy to deal with. So let's do it this first.

[00:10:23]We have one from one to uh let's actually let's do this. This is this would be easier. Three halves.

[00:10:34]Yeah. Three halves, right?

[00:10:37]Then this is going to be two.

[00:10:40]and then x2 x. Okay, so we're just going to constructively do this. So it's going to split at one to three halves and then after after three halves three halves to two then that means the bottom is now what? Three, right? So I put three halves * 2 is equal to three.

[00:11:05]So, so we're just going to just kind of slowly deconstruct this by hand.

[00:11:16]Okay, good grievance. All right, now to for the nasty part, splitting it into roots.

[00:11:24]So, is there a root that we have to split between one and three halves? The answer is yes, roo<unk> two. And the reason is because uh square root of two is like 1.4.

[00:11:37]Um it's hard to kind of know that. So because of that, this is going to be like what? One. This is going to be two.

[00:11:49]So it's going to be like this.

[00:11:55]Okay. But then here at root two, we have what? Three halves. 1.5.

[00:12:05]And so now because this is like <unk>2 what's what's <unk>2 squared two, right?

[00:12:12]So now this is like uh three. Okay. Be careful cuz that's a ceiling function, not a floor function.

[00:12:19]So you got to be careful with that. Is there anything between three halves and two? Yes. Square root of three. Square root of three is like 1.7 something. And so that's going to be like what?

[00:12:32]Um yeah, three three. So also just a fraction part and then square root of three from two.

[00:12:43]It's going to we're going to get four fractional apart of this.

[00:12:49]Oh god. I mean what's the fraction apart between one and roo<unk>2? It's just like is it just one?

[00:12:58]I think it's just one, right?

[00:13:03]Isn't that Oh, I'm dumb. I'm so dumb.

[00:13:06]Uh, just just use this. Don't over complicate it. Don't over complicate it.

[00:13:10]Just just use this. Think of it as that.

[00:13:14]Okay. So, this is like x - one. Will not be lazy. Will not be lazy.

[00:13:21]Got it.

[00:13:24]I just put it here. All right.

[00:13:27]So we have So the indefinite integral was like x^2 / 2 - x. I would not be able to do this in 3 minutes. Oh my god, this takes forever.

[00:13:41]Oh god. 9 over 4 9 over 8. Oh my god. I don't want to do this.

[00:13:49]Oh jeez. Why? Why would you do this to your competitors?

[00:13:56]It's arithmetic time. All right. 1 minus <unk>2.

[00:14:01]What the hell? Why would you make such a problem?

[00:14:06]Just leave it as it is. Just leave it as it is. I don't want to deal with this.

[00:14:11]Why? Why would you? Why? Why? Rahul.

[00:14:16]Rahul. You son of a Are you the same person that made that logorithm that evil logorithm problem?

[00:14:24]You are. Of course. Of course. God damn you, Rahul. Why? Why would you do such a thing? Why would you give me such a thing?

[00:14:35]This is actually evil.

[00:14:38]Um, how did you do it? Okay, so I split it up exactly like that. Yeah, this is disgusting.

[00:14:48]Oh, wow. It simplifies. This whole thing simplifies to this. I refuse to believe that. I refuse to believe that. Oh, is that what they have? Oh, yeah, that is exactly what they have.

[00:15:04]Okay. Huh.

[00:15:07]Okay. I lied. Perfect. Okay. Yeah. Yeah.

[00:15:10]So, that that's correct. And I was just I'm just too lazy. I don't want to do this part. Uh but now now I'm curious.

[00:15:17]16 16 - 27.

[00:15:22]How does that simplify 2/3 into 2/3? Uh 27 - 16 is 11.

[00:15:31]11.

[00:15:36]So how does this simplify to something nice?

[00:15:39]Because this it it does not look like it would does not look like it would. Uh, -3 16 time 8. I don't know. 128.

[00:15:53]I I don't know if that's correct. I'm just going to I'm too lazy to check.

[00:15:58]95.

[00:16:02]I have no idea.

[00:16:05]I have I have no idea. I don't know. My fractions are wrong somewhere. Somewhere wrong. Somewhere's somewhere is wrong with with the fraction part. Um yeah, I don't know.

[00:16:19]I don't know. I don't know. 11 over 16.

[00:16:22]That sounds like something I had.

[00:16:26]11. Oh, okay. So, there's So, this is extra then. So, I have an extra 8/3 somewhere. I don't know where I got that from. I don't know. I don't know. This is bashy. This is nasty.

[00:16:42]I don't know.

[00:16:45]Oh god, you have to manually deconstruct it. That's so awful.

[00:16:50]And you only have like three minutes.

[00:16:55]Nasty. Next integral. Oh, that's it.

[00:16:59]Okay, round of four. Oh, boy. I think this is where quarterfinals hits. All right. Well, there you go. Uh, I'm sorry.

[00:17:10]Oh god, that's terrible. This so bashy.

[00:17:15]It's getting ba I feel like it's getting bashier. Yeah, I don't want to attempt this. My condolences to whoever got this. This is so unfair. This is disgusting.

[00:17:26]Like this is how you give Indians PTSD from Gain.

[00:17:31]Like this is awful. Like come on, don't do this.

[00:17:36]And then this was nice.

[00:17:39]And and this was also nice. Uh this was slick too. Squareepsky for the win.

[00:17:45]Squareepsky. Thank you Aush for that.

[00:17:48]But Oh yeah. And and the fact that it's nice for Don't do this. Don't Don't do this. But the the you know the integration by parts that I did for the intermediates, you know, it's surprisingly came out nicely. A lot nicer than I expected.

[00:18:06]But yeah, this is just what the hell?

[00:18:11]This is just cruelty. This is evil. This is really evil. And then speed bash.

[00:18:18]Okay, so you know the drill, right? Stay tuned for the next part. Uh, thank you so much for watching and I'll see you guys in the next video. See you.