00:01
Let's evaluate this integral by considering several cases for k.
00:05
So here are three cases that will cover every single case out there.
00:10
Case 1, k0, case 2, positive k, and case 3, negative k.
00:17
So in case 1, we can rewrite this integral as just 1 over x squared, and then use the power rule to evaluate that integral.
00:29
So that would be the answer for k equals 0.
00:33
Now let's consider the case when k is positive.
00:39
So then we have 1 over x squared plus k.
00:47
We can rewrite that as x squared plus radical k squared, and then we can use the tricks up here.
00:56
Here we should take x to be radical k, tan theta, then d x, radical k, secan squared d theta, and then this integral can be written as radical k, secan squared.
01:21
So that's a k over here.
01:24
And then on bottom, when we square x, we get k times tangent squared and then plus this k here.
01:34
So let me go ahead and factor out that k.
01:38
And then we use the fact that tan squared plus 1 is equal to secan squared, pull off the constant.
01:48
And then now, let's go to the next page, since i'm running out of room here.
01:56
We have root k over k times integral of d theta, which is just theta.
02:02
And then using our trig sub over here, x equals root k tangent, we could solve that for theta...