I Tried University of Texas Austin Integration Bee 2025 Quarterfinals

Added: 2026-05-09

[00:00:00]What's up you guys? Silver here and we're going to try some integrals from University of Texas. This is the 2025 integration B. I have not tried or attempted these. I probably should not scroll down. I kind of want to keep it more of a surprise, but um but I just kind of have this PDF just kind of sitting on my files, but I've never attempted or tried any of these integrals yet cuz I just I don't know. I liked site reading them. I think just site reading integrals. This is very fun. So, uh I don't have I don't think I have the qualifying stuff. I think I only have just the quarterfinals and uh the furthers. So, let me go ahead.

[00:00:46]Let's start with the first quarterfinals.

[00:00:50]Immediately I see GCD and I'm like, "Oh my goodness, what have I done?" Uh I fun fact I I started the tradition of adding GCD and LCMS and integrals and uh I I might I might regret it um just cuz I hate peacewise integrals but yeah I I started the tradition of GCDs instead of an integral. This was like UC Berkeley time like holy hell like years.

[00:01:21]Oh my god this is going to be a H okay. I probably should not list it like this. I think the best way to list it is through intervals. So for E, E squ, E cubed, and E6.

[00:01:40]Now for this interval I'm it's definitely from one to two um from e E squares um two to three yeah 3 to 4 and then oops four to four to five cuz the GC of four and six is two. Oh my god.

[00:02:12]Why? Why? Why would you do this?

[00:02:16]Is there Is there like a faster way? I have to do this all the way up to 2025.

[00:02:25]Yo, this is some comnatorics type stuff.

[00:02:29]Oh my god. Okay.

[00:02:34]Ah ah so we have to so how many e where we have um the gcd n = to 1. Oh my god. Why would you turn this into a cominatorics integral?

[00:02:59]No.

[00:03:01]This is awful.

[00:03:03]113.

[00:03:05]Here I think we'll we'll um what's it called? Not over counting. Uh counter counting.

[00:03:13]Is that what it's called? Counter counting. I don't think that's what it's called. That sounds dumb. I don't know why I called it counter. Complimentary counting. That's what it's called. We We'll save this for complimentary counting. And then so there's 2025.

[00:03:28]We'll split it in 2020, right? 22.

[00:03:32]Oh, no. It'll be 2024.

[00:03:35]It'll be 2024 sections. I got to be careful. We'll split it in 2024 intervals. There's 2024 intervals, right? There's nothing 2025 and like what? Like what would it be here? Cuz we stopped at our upper bounds 2025. So, it would be 2024 intervals. How many is divisible by six? Oh my god. I cannot believe you're making me do cominatorics. I hate comnatorics.

[00:04:00]There's I just I just have no patience for comics. Uh is that right? Seven.

[00:04:08]So there's 337 intervals. Oh, I got to be careful.

[00:04:16]Crap. Yeah, I got to be careful cuz I have to do like this, but then I have to like half I might have to half it cuz half of the divisib the half of the integers that are divisible by three are divisible by six. So, I'm sorry. Yeah, half of multiples of three are multiples of six. I think that's the right way to word it. But um but you you get the idea, right? I hate this. So do you. This is not what I'm used to. That means that if there are if there's 674 of these, then there are um then there are 330.

[00:05:04]Oh, wow. Exactly 237.

[00:05:07]Okay.

[00:05:09]What am I saying? I mean, that actually makes a lot more sense. If if you're going to have multiples of six, half of them are no half of multiples of three are multiples of six. What am I doing? I I should have known this trivial trivially.

[00:05:23]God damn it. Combinatorics always gets me. Okay.

[00:05:27]E square. Oh my god. And then I have to count the even numbers and I have to make sure I don't over count for the freaking multiples of six. God dang it.

[00:05:41]Okay, so there's 20 24 intervals. Yes, there's there's 2012 intervals with even numbers. But then what about then I have to subtract that subtract um integers that are multiples of six, which is 337.

[00:06:02]God, what am I do? What are you making me do?

[00:06:06]Like this is this is insanity. This is quarterfinals. This is so evil. This is so bashy for quarterfinals.

[00:06:15]Oh my god.

[00:06:17]And then I have to subtract all of this.

[00:06:21]Um, this was 674, right? Wow.

[00:06:27]Oh my god. What am I doing? This This is obvious as well, right? If if half of your if half of your integers are even, then of course half of your integers are odd. I'm so I cannot with this.

[00:06:40]I Oh my god. This is why I can never be a math Olympian.

[00:06:46]Okay. So what we get?

[00:06:51]So I think our answer is just this plus 337 e cub + 675 e^² + 675 e. And I believe that is our I don't know. I now I want to check. I kind of want to see what the answer is. Does it want to let me know?

[00:07:13]Uh, did I do something wrong?

[00:07:19]I definitely did something wrong. I don't know. I have no idea.

[00:07:24]I did something wrong. I don't know.

[00:07:28]There is There's I've I counted something. There's I don't Oh, Lord. Are you serious?

[00:07:38]the GCD before. Yeah, I don't know. This is awful. I don't know. I I don't know.

[00:07:46]I don't know.

[00:07:48]Just leave. Just leave. Just take me to the next integral. I'm done with this. I don't No more peacewise. No more cominatorics integrals, please.

[00:07:58]I I'm done. I'm so done with this.

[00:08:02]I'm so mad. But I can't even I can't even believe the fact that it it was that easy to count. Like it makes sense that half of the multiples of six are multiples of three. Half the integers are even. Half the integers are obviously odd. Like why did I why on earth did I waste my time counting?

[00:08:22]God gez. Okay, let's just go. Let's just go. All right.

[00:08:28]U= 1 /x. I think we kind of already see the the correlation with this, right?

[00:08:34]Okay. Super easy. Why is that in quart like why can't we Oh my god. Like at least that was an easy quarterfinals.

[00:08:41]This was You just have to have very good sense of number theory and comics I guess. I don't know. I don't do quant. I don't I have a life you guys. All right.

[00:08:52]So, this one, this is kind of terrifying me. That 2025 is so unnecessary.

[00:08:58]But maybe it's a distraction. I don't know. Let's call that uh a for now. If I Oh, what am I doing? It's not cosine square. What am I doing? It's not It's cosine of x. Oh my god. I'm like, something ain't right. But now, now I see it. But u= 1 + s of x, right? Then you this is much nicer. Now you could just do let wal of uh my god I don't know what happened there. Let w= square root of u.

[00:09:31]Okay. And then you get 2du uh dw um a + uh w^2 - 1.

[00:09:41]This is as as much as this looks how awkward this looks. This is inverse uh inverse tangent integral. What a problem shock. Wow, that definitely problem shocked me. Uh, but that's cuz my my dumbness. Let me know in the comments if I did something wrong or if that's incorrect. Um, let me know or make a video out of it and I'll uh I'll pin it if you send the link to your video most likely. Okay, so match two is uh very terrifying. I don't know whether this is going to be this. I don't know. This This looks sus.

[00:10:25]Yeah, let's turn this. Let's Let's just put this in exponential form cuz this is this is scaring me. Um e to the x - e^x.

[00:10:33]Wait a minute.

[00:10:35]Ain't no way, chief.

[00:10:39]This would be like minus 2.

[00:10:44]Wait, no. Wait, hold on. E to X - EX^2.

[00:10:51]Is this not what it gives me? E to X 2X - 2 + E -2X - 4.

[00:11:02]Um, this gives me -6.

[00:11:09]Okay, I had my hopes up, but this is Oh no.

[00:11:19]Oh no.

[00:11:22]Don't do this. Do I have to do the complete the square thing? E to the X.

[00:11:28]You know what? Screw it. Can I just go like Let me know in the comments if I did this incorrectly.

[00:11:34]Um, I hope I hope I don't need the one over 8. I don't think I do. Um, okay.

[00:11:44]Oh lord.

[00:11:47]Oh god, this is getting bashy. Like these are like bashy integrals so far.

[00:11:53]Um, what on earth? Huh? Wouldn't this equal to zero? It's not zero.

[00:12:01]Huh?

[00:12:03]Is this pi square over 6? No, this this is like pi square over 3.

[00:12:11]Huh?

[00:12:12]It's not zero. Wait, but this isn't this equal to ln of Oh, ln of 2 tangent of x.

[00:12:20]I'm a d. Oh my god. This is equal to pi / 2 ln of two.

[00:12:27]Yes, it does. Oh my god, I figured it out. But yes, ln of tangent of x the integral of ln of tangent of x from 0 to p<unk> / 2 equals to zero. But I forgot you get a uh this gives you a a two left over. So you get like two tangent of x. So the tangent of x part is zero. ln of two. You just get ln of two. So you the answer is pi /2 * ln of 2. Okay. God, I can't believe I missed that. Don't rush like me. This is um glass is this is glasses master theorem.

[00:13:07]Oh no, no, no. I got to be careful. I got to be careful. Oh, I almost I I almost done goofed myself. You got to be very careful. Um it's not you you got to do a little bit more than glasses master theorem. This this is this is not this is not this.

[00:13:31]So you you got to be you got to be very careful. Can I just go like this? I think that should be fine.

[00:13:40]So I believe I could just go like 1 / e^2 and this is just e to the x - 1x^2 dx and that should be good. So now this just gives me roo<unk> of pi over e^2 by glasses master theorem. I think that's correct. I feel like I did something illegal. I'm second guessing myself a lot. So, I'm just gonna just don't mind me checking.

[00:14:12]Oh, no. No, that's uh negative infinity to infinity. Um, yeah. Okay. Awesome.

[00:14:21]Awesome. Nice. Nice integral. It's a really nice glasses master theorem practice and a good use of square square technique, too. Oh, come on. Don't do this. No. Ew.

[00:14:35]I don't even want to attempt this. I don't even know how to do this. Like things like this. Like if you're going to give me something like this, like I don't know. I don't graph. I don't graph for a living.

[00:14:49]Why is it like this? Oh god. MIT, look what you influenced. The way I kind of see this.

[00:14:57]You know what? I take that back. I don't see anything. I don't know how to do this integral. I the minimum I have an idea.

[00:15:05]I probably I think I just needed to kind of deconstruct this. Pretty much what this is saying is the min of this is pretty much natural numbers. I don't know why they wrote it like this. Uh I don't know. Oh god. Uh this is not fractional part. Just just so you guys know. This is just how I write like minimum maximum stuff. It's just my notation. Uh my I'm the notation I'm used to. But the oh god the minimum I don't know I know for sure from one to two the minimum is literally this and then for two to three it would be like I don't know what's the minimum what would be the minimum 3 factorial 6 over 8 oh okay maybe this is the minimum I Yes. But is there really a pattern to this? Is there really a pattern to this?

[00:16:07]2 over 9.

[00:16:12]This actually looks smaller so far. That looks like a minimum. Three.

[00:16:20]That's one over. Oh, never mind. This is more of a minimum. But then it it will get bigger eventually, right? Because when we get to like bigger factorial numbers, then we'll get like two, three, four, five, six, 7, 8, nine, and whatever. And it's going to be like way bigger, right? Well, even like if it was like for five, like yeah, like four integers are going to be less than five, but then like we'll have more integers later on that's bigger than five, you know? So of course like along the way here it's going to be like completely big. So of course the minimum is always going to be somewhere in the first few terms.

[00:17:06]Oh my god. But I don't I don't see a pattern here. What is this?

[00:17:13]Why you do this? Oh my god.

[00:17:16]This is so awful. I don't I don't even know how to even How do I even go about this? 4 over 32. Oh, wait. This might be the minimum. Is this the minimum?

[00:17:29]Wait.

[00:17:30]Huh? Wait. So, the third one, three, the third term. Four is the fourth term.

[00:17:38]That's minimum. Huh?

[00:17:41]Is that what this pattern is going? And then five, six is the smallest. The minimum is going to be the fifth term like this.

[00:17:55]Is that what this is?

[00:17:58]Is this what you're showing me? Okay. I I think I kind of I'm starting to kind of get the gist of this. The minimum is always going to be like this is the pattern I'm seeing, but I don't know if this is correct. But let's just kind of Oh, no. This should be N. Huh?

[00:18:23]But we don't even know what this is.

[00:18:25]There's no closed form of this.

[00:18:28]I'm so lost. I have Oh, I forgot about this.

[00:18:35]I forgot about this. Oh, that's what that ceiling function was for. So, there is no closed form. Oh, dear Lord.

[00:18:47]uh 1 + 2 factorial.

[00:18:52]Oh, I mean that's literally Okay, I'm literally doing what I just wrote down.

[00:18:56]1 9th + 3 over 32 plus um 120 of I have no idea 625. We know for sure this is like 1.5 plus 0.1 one plus a bunch of nonsense. This is like something I don't know. I'm going to go with two. If it's three, I'm going be so mad. But I'm just going to say it equals to two. I have no idea. That was that's that's an evil. We're still in quarterfinals, just so you guys know. I would not be able to do this in time.

[00:19:45]Okay. Why Why would you do this? You're going to bash us? Come on. If I let u= 1 /x, I get something like u cub 1 + u ^2. That's this one's going to be taken care of. But then I'll have x.

[00:20:09]Oh, sadly this is not a if it was from zero to infinity. Oh, this would have been beautiful. Would have been beautiful. But I'll just I'll I'll I won't over complicate it. I'll just do it. I'm just very lazy. Okay. If I plug in negative 1, I get two 1 here. If you see like x like that like 8x 8 8x like this immediately let x equal to zero immediately right when you let x equal to 0 then 0 is equal to - 1/2 plus what b so b is equal to 12 now you just have to find a so now I will let x equal to 1 and I know I'm safe cuz I've never let x equal to 1. It's I let x go negative 1 before to find this. So x goal to 1.

[00:21:07]This gives me 1 = uh 21 + a + 12.

[00:21:18]This gives me * 2. So this would be like 2 a + 1.

[00:21:23]Oh, would you look at that? 2 a + one.

[00:21:25]This one and this negative 1 cancels out. So, a is equal to 12. Awesome. I'm too lazy to simplify my my answer. Uh, there you go. That's I'm just going to leave it like that.

[00:21:38]Oh god, what a bashy integral. Okay.

[00:21:43]Arc cosine. You know what's easier than integrating inverse cosine of cube root?

[00:21:50]Cosine cube.

[00:21:53]Yes. Yes. One over two. Oh, do not do this to me. I just noticed what the lower bound is.

[00:22:04]You're going to Are you really going to like block me with this?

[00:22:11]Oh, great heavens. Yeah, they are going to block me with it. Oh my god.

[00:22:16]Come on. Why you you got to ruin the fun with this?

[00:22:23]No. You know what? Screw you. I I'm plugging it in. I'm going to leave it like this if this is how you're going to be. 1 over cube root of 2<unk>2.

[00:22:34]Yeah. How you like them apples? There you go. Okay, man. I don't want to do this. This is nasty. This Why? Why over 2<unk>2?

[00:22:44]Why one over two <unk>2? Who the hell?

[00:22:47]Who wrote this integral? Seeant square l of secant square h integration integration by parts x.

[00:22:56]This is too easy.

[00:23:00]It's too easy. Too easy.

[00:23:04]Okay, that was the easiest quarterfinals ever.

[00:23:08]What the hell? Oh, all right. We're going Norwegian. Is this even a nor is this even a Norwegian substitution thing? I don't know if this does this even Oh, yes it is. Yes, it is. Yes, it is. I had to look closely. It's I didn't It's It's squared. Yeah, this this does work.

[00:23:32]Yep. Uh x over e 2x + 1. Yeah. Uh if you let u equ= x over e^x this du does equal to this at least to my knowledge assumingly.

[00:23:53]So then this is just going to equal to inverse tangent of uh xx e to thex plus c.

[00:24:02]I wouldn't say popular. I I take it back. It's not a popular integral, but if you're very experienced, you you will see that right away. I know for sure like for those who study G, like you the J exam, like you you'll come across that quite commonly. This looks legit.

[00:24:23]Um 1 + 4x - 1 2x.

[00:24:33]Yep. So 12 of<unk> 1 + 4x - 1 dx. This looks legit. All right, that was a satisfying integral.

[00:24:49]And finally to semifinals. Okay. Oh god, just looking at this already. Oh god.

[00:24:58]Um, I might have to Yeah, I'll I'll put this in a separate video. So, that was um that was quarterfinals. A lot of I wouldn't say a lot. Half was Bashy.

[00:25:12]Half was kind of a good scare. Um, God, this was a This was just evil. This was a good scare. And this was just evil. Um, and this was a good scare. All right. So, yep. That's all for Oh, yeah.

[00:25:28]And this is just that this is just rude.

[00:25:31]This lower bound was just rude. Um yeah, match four. Whoever got match four, y'all are lucky, man. Whoever's the competitors for match four is so unfair. You guys got the nicest quarterfinal integrals. All right. Well, that's for quarterfinals.

[00:25:49]Uh I'll do the semifinals and the finals next in the future videos. All right. So stay tuned for that. Thank you for watching and I'll see you guys in the next video.