00:01
Okay, because we have a plus sign here, we know it's a tangent substitution.
00:06
So what i'm going to do is x squared equals 4 tangent squared theta, which is the substitution you gave me, x equals 2 tangent theta.
00:17
But i always think of it like this, because then when i calculate this, x squared plus 4, that is 4 tangent squared theta plus 4.
00:29
That is 4 tangent squared theta plus 1.
00:38
Tangent squared theta plus 1 is secant squared theta.
00:44
So that is 2 secant theta.
00:49
And then from this, dx is 2 secant theta d theta.
01:02
Okay.
01:04
So now let's make the substitution.
01:06
In place of x cubed, i get 2 tangent theta, cubed.
01:14
In place of the square root of x squared plus four, i get two secant theta.
01:20
In place of dx, i get two second theta, d theta.
01:26
Okay, those cancel.
01:29
I get eight, tangent cubed theta, d theta.
01:37
Okay, because it's an odd power of tangent, i break it up into an even, and the leftovers.
01:48
Okay, remember the tangent squared of theta is secant squared theta minus one.
02:08
Okay, so i'm going to make this two integrals.
02:12
I'm going to make it the first one, which is tangent times secant squared, but i'm going to put the tangent first tangent.
02:18
Sequent squared theta d theta minus eight tangent theta d theta.
02:29
Okay, so on this one, i'm going to let you be the tangent of theta, then d, d, and, is secant squared theta d theta.
02:41
So this is eight integral u -d -u.
02:45
That's eight, u squared over two.
02:51
That's four, tangent squared theta.
02:56
Okay, we'll come back and fix it in a minute.
03:00
Okay, on this one, oops, down here, i'm going to break the tangent up, i mean change it to sign over cosine.
03:13
So in this one, i'm going to let you be the cosine of theta, and then du is minus sine theta d theta...