I Tried University of Texas Austin Integration Bee 2025 Semifinals

Added: 2026-05-11

[00:00:00]Oh, sorry you guys. We're going to come back to the University of Texas Integration B 2025.

[00:00:07]Coming back with semi-finals and finals.

[00:00:10]Depending depending how long this video takes. But uh let's get started to semi-finals as we continued. And um just look at this first integral.

[00:00:24]>> [laughter] >> Uh let me uh let me just kind of screenshot this.

[00:00:29]First let's let's first simplify that.

[00:00:32]Right. Start small. Then we just power of K equal to Oh, well, K + 1 equal to tangent X.

[00:00:40]Interesting. So, this whole thing Well, Oh. Oh.

[00:00:47]Okay. 2025 K equals to 1. Okay, so this is it's it's a um It's the Queen's rule. It's a famous Queen's rule.

[00:01:01]Wow, what a nice problem shock.

[00:01:04]That's a very nice problem shock. This is just simply pi over four.

[00:01:09]However many times 2025.

[00:01:12]Yeah.

[00:01:14]This is just 2025 pi over four.

[00:01:18]I think that's the correct answer.

[00:01:21]Because this whole this whole integral here this just equals pi over four.

[00:01:27]So, if I iterate Yeah, yeah, I know that that should That looks correct. Yeah, 2025 pi over four. Yeah. That That'll be my answer. That's actually a nice problem shock. Um very nice problem shock. There's actually there's there's something I like to do on this part. You can do integration by parts.

[00:01:45]But I like to just for my mental sake I just let you equal square root of X + 2025.

[00:01:55]So, instead of just doing integration by parts, I just do it like this.

[00:01:59]And that way, I just know immediately, like just immediately get to the answer.

[00:02:06]Well, that should be 2 U + C. That should be my answer.

[00:02:10]Okay.

[00:02:11]Is it me or the semi-finals just a bit easier than the quarter-finals?

[00:02:18]I think I For some reason, the semi-finals are a bit easy. Maybe just just for match one.

[00:02:24]Just for match one. Oh god. Is this caveman? This might be caveman.

[00:02:30]Just testing it. Right, if I take X4 out, I get X to power of 5 1 + X to power of -4.

[00:02:40]Ah.

[00:02:41]Caveman. Yeah, these These should have been quarter-finals, but um it's okay. Very good.

[00:02:49]Very good.

[00:02:51]Too easy. Match one was too easy.

[00:02:55]Too easy. Oh my goodness. What is this?

[00:02:59]That's not even fair. Why is match one easier than match two?

[00:03:03]Ah, my condolences to these competitors.

[00:03:07]Okay.

[00:03:09]Oh my god.

[00:03:10]>> [laughter] >> What the hell?

[00:03:13]Who Who sorted these integrals?

[00:03:20]Oh my god. This is so much harder.

[00:03:23]So, this integral is actually tricky. If you're If you've never seen this, um or maybe it's just me. Let me first check cuz this is like 2 SC.

[00:03:35]If I do like secant square here, then this is going to be like tangent X secant square tangent Ah.

[00:03:48]See, if you were advanced, if you're a bit too advanced, you would have done like something like the sine 2x + 1 trick, but no, it's none of that. It's you got to you got to keep your beginner mindset sometimes. But that's that's a bit tricky to do.

[00:04:05]Uh because we're so used to certain advanced techniques.

[00:04:09]But yeah, this is just um yeah, letting you equal tangent of x. I hope I did something right. I hope that's correct. If not, let me know in the comments and correct me or make a video out of it.

[00:04:25]Um and I will I will pin it. All right, you know you know the drill.

[00:04:30]To all my advanced audiences, you know the drill. Let's go.

[00:04:36]This is n + 1 and n + 1 we get x + n x - n x n square dx.

[00:04:52]Okay.

[00:04:54]Awesome.

[00:04:55]Oh, wait, what am I doing? Oh.

[00:04:59]Oh my god. [laughter] What am I doing? Oh, I think there's a faster Wait, actually, maybe maybe this is okay. You can you can continue on.

[00:05:09]You can continue on. I'll I have a second path.

[00:05:12]What if I keep forgetting um I can go like this.

[00:05:18]Um like that.

[00:05:22]So, this is this would be uh I don't know why this is used x.

[00:05:28]Um but if I follow that route, we you let you equal x minus n then this is like what?

[00:05:41]From zero to one u plus n plus un all over U plus N and square. I don't think it makes any easier. I think it's I think I just kind of screwed it up. Or maybe it doesn't make really it doesn't really make a difference. I I see no difference.

[00:06:02]This is 1/X 1/N square plus X minus N XN. Yeah, it doesn't seem to be a difference.

[00:06:13]The only The only difference well Yeah, the only slight difference is like the balance are different. So, it's I guess it's easier to compute here.

[00:06:21]I guess I guess this is a bit easier a much faster way to mentally see it.

[00:06:28]I guess. I guess.

[00:06:31]Pi square plus This is like 1/N minus U plus N So, that's like This is going to bring like DU Oh, what am I doing?

[00:06:58]Oh, no no no no no no. I'm that's this is N not U. So, this [snorts] would be like 1/N. Oh, good grief. I can't integrate. Uh this 1/N though, where do I how do I get rid of this?

[00:07:13]I must have done something wrong.

[00:07:15]I'm pretty sure I must have done something wrong. I have no idea what went wrong here. Let me let's not go through this path.

[00:07:22]I've I've hallucinated myself into something.

[00:07:26]Um but let's let's keep it simple, right?

[00:07:30]So, let's not overcomplicate. Let's keep it simple.

[00:07:33]N equals to 1 and N plus 1 1/N square. This is uh 1/N minus 1/X Um Huh? I get some I get the same thing.

[00:07:51]Oh, please tell me please tell me it's not one of those macaroni BS.

[00:07:57]I swear to God if this is one of those if it's one of those um this I swear to God and this like the telescoping thingy.

[00:08:17]Um Like this and it's like um What? If I plug in one, right? Ln of zero and then this goes to infinity.

[00:08:33]Right? Ln of K.

[00:08:37]Oh my God.

[00:08:39]Please tell me this isn't this BS.

[00:08:46]Let's not have some macaroni into this integration bee. Please.

[00:08:51]Please. I'm I'm actually curious.

[00:08:55]Of of course. Of course Wolfram Wolfram Alpha can't they can't compute. Oh my goodness. Oh my God.

[00:09:06]Come on.

[00:09:08]Really?

[00:09:11]Oh my God.

[00:09:14]It's so unfair. I apologize to the audience that I faced with you with I I didn't think there's going to be a macaroni constant in this integral. I didn't I I do not have that in my training. I don't usually teach that because that's not really you don't really see that in integration bees and I hope If stays like that.

[00:09:36]But I Is that a double integral? Oh my god.

[00:09:41]I'm too afraid to do this one.

[00:09:44]I genuinely do not want to try this.

[00:09:47]That That looks disgusting.

[00:09:50]What What are you trying to do?

[00:09:53]Don't know if it's going to make it any better. At least this is a constant. So we can kind of move that to the front.

[00:10:01]Um I don't know what I'm doing here. I'm just kind of playing around. Oh, I guess I could divide four top and bottom.

[00:10:11]Now I think about it. What do you want me to do with this? This is so nasty. Oh my god.

[00:10:19]Come on.

[00:10:21]I Do I have to turn these into signs and cosines?

[00:10:26]Is that what this is?

[00:10:27]Is it one of those G-Man type linear algebra BS?

[00:10:31]Oh my god. Cosine.

[00:10:34]Cosine.

[00:10:35]This is not a motivating integral.

[00:10:38]Oh my god.

[00:10:42]Doesn't even work at all.

[00:10:47]This is so unmotivating. All right.

[00:10:51]I have no idea.

[00:10:53]I I'm not getting anywhere. This It's some sort of math Olympian trig identity type crap. I don't know. This is I have no idea.

[00:11:04]I have no idea. I don't know a faster way.

[00:11:09]I don't know. It's some sort of trig identity.

[00:11:13]Uh Good grief. I Man, this Whoever had this match two, I'm so sorry for you. Match one is so Match one is easier than quarterfinals entirely. It's so weird. I think the video's long enough, surprisingly.

[00:11:29]Um Yeah, I'll make grand finals for the next video. So, stay tuned for that.

[00:11:37]Thank you so much for watching and I'll see you guys in the next part.