00:06
Okay, so we're given the following integral.
00:11
That should actually be a minus.
00:17
And we can rewrite it as the following integral.
00:22
So that's 4 minus 1 minus 2x, minus x squared, dx.
00:28
And this whole thing is if you factor out the negative 1, 4 minus 1 plus x all squared.
00:38
So we have the integral of the square root of that.
00:41
And so what is that equal to? so that sort of looks like we could bring in a sine trick sum.
00:53
So our hope is that we have 4 minus 4 sine squared theta under the radical.
01:04
So we could say that that is equal to, so it would be equal to 4 cosine squared, which would then be equal to just 2 cosine.
01:23
So if we want that to happen, what do we have to do? we have to let 1 plus x equal the square root of this.
01:32
So 2, sine theta.
01:36
And so dx is equal to 2.
01:39
No, cosine theta, d theta.
01:44
And so we already reduced this.
01:47
So we determined that that would be equal to 2 cosine theta.
01:53
So we'd have the integral.
01:56
And now we'd have times 2 cosine theta d theta.
02:02
And so just move that.
02:05
And so now we'd have 4 times the integral of cosine squared theta d theta.
02:17
Now we can rewrite this as 1 plus cosine of 2 theta d theta, multiply by a one -half on the outside.
02:32
And so you're going to have four times that.
02:36
So that's going to cancel and that's going to make that a two.
02:42
Let's just erase it so it's not a sloppy.
02:49
And all right.
02:52
So we're almost there now.
02:53
This is easy to integrate.
02:56
So we just get 2 times theta plus sine of 2 theta...