00:01
Hello, and here we're asked to integrate 2 times the log base 2 of x minus 1 or x minus 1 from 2 to 3.
00:07
So we're first going to start by factoring out the constants.
00:11
So we'll have a 2 move to the outside.
00:15
And then we can do a nice log substitution using the change of base rule.
00:21
So we will have log base 2 of x minus 1 equal to ln of x minus 1, over ln2.
00:32
This is just the standard change of base.
00:35
All right, and we can substitute that in, and we will get 2 over ln of 2 times the integral from 2 to 3 times ln x minus 1 over x minus 1 dx.
00:52
Great.
00:54
From here, we can do two u substitutions, just for clarity.
01:01
Let's get rid of this.
01:03
So first we'll set u equal to x minus 1, and we'll have to change the base here, or change the bounds here.
01:13
When the bounds are evaluated here, we'll get a bound of 1 and 2, because we're just subtracting 1.
01:19
So we'll get this as equal to 2 for ln2 times the integral from 1 to 2 of ln, u over u, du, and yes, the du will stay the same as dx, because du is equal to dx here.
01:37
Just taking the derivative real quick.
01:42
Oops, just look at that.
01:44
Great...