What Math Olympiads practice? #mathematics #calculus #integralcalculus #competitionmath

Added: 2026-05-06

[00:00:00]You know, it's been a long time where we have to solve an integral by manipulating the root. We're working with the integral of x ^ 5 or<unk> of 1 - x cub with some disrespect to x. And this was surprisingly a practice problem for math Olympian students. And I'm talking this had to be a very basic problem. But let me show you how it works. I talk about this in book of integrals where we have page 24 manipulating the root. And the idea is to make a substitution, manipulate the x values, and then solve for anything else that is going to have a u. Okay, so here we go. We always want to make our u value equal to the stuff inside the square root. We have 1 - x cubed. And now we have du is equal to -x^2 or -3x^2 dx. Let's go ahead and move this to the other side. So that we have du over -3x^2 is equal to dx. And this will be our differential. So now let's go ahead and take care of this. We have the integral of x ^ 5. This will turn into u to the 1/2 times and then we have our dx which is this entire [music] thing. So du over -3x^2. So what's going to happen? Well the x cub and the x^2 on the bottom here will turn into an x cub.

[00:01:06]We'll talk about that in a second. And this -3 will go on the outside. So we have - 1/3 integral of x3r u to the 12 du. And the issue is that now we have an integral involving x and u. So, what we're going to do is revisit this and we're going to manipulate this. We're going to go ahead and move this to the other side. Subtract the u. So, we have x cub is equal to 1 minus u. And that is exactly what we want cuz look, we have the x cub. So, now we have - 1/3 integral of 1 - u u to the 1/2 du. Let's go ahead and um multiply or distribute this into both terms. U to the 1/2 minus u to the 3s du. And now we get to actually integrating. This will be u to three * 2/3 minus 2s u to 5 plus your cheese. But let's not forget that u was equal to this over here 1 - x cub. So this will become -2 over 9 1 - x [music] cub to the power of 3 + 2 over 15 1 - x cub to the power of 5 halves plus your cheese. And guys, we are officially done integrating something where we just have to manipulate some values.