00:01
Evaluate the following two integrals.
00:03
The first is the integral of 2 plus 5x times the sign of 2x dx.
00:17
So we're going to separate this into two integrals to make things easier.
00:23
So we have the integral of 2 times the sign of 2x dx plus 5x times the sine, the sine, of 5x times the sign of 2x dx.
00:41
Let's focus on this first integral.
00:55
We can integrate this via a change of variable.
01:01
So let's pose that u is equal to 2x, such that du is 2dx.
01:08
So we then find that our integral becomes the integral of the sign of u, and the integral of a sign is minus a cosine.
01:32
Let's focus on our second integral.
01:35
So this factor of 5, i'll just put it to the front and focus on just x times the sign of 2x dx.
01:50
We're going to also pose a change a variable.
01:54
Well really we're going to apply in direction my parts.
01:57
So let's pose u equal to x such that du becomes d x.
02:02
We're also going to need dv.
02:05
Dv in this case will be the remainder of our terms in our integral.
02:09
That is the sign of 2x dx.
02:13
We need to integrate this to find v, and the integral of sine of 2x is minus, we just found it before.
02:20
It's minus the cost of 2x divided by 2.
02:27
So this integral will give us minus x times the cost of 2x divided by 2.
02:37
Plus the integral of the cost of 2x divided by 2 the x.
02:46
The integral of the cost of 2x gives us the sign of 2x divided by 2.
02:52
So we now find minus x times the cost of 2x divided by 2 plus the sine of 2x divided by 4.
03:10
Now let's put everything together.
03:14
Our integral will give us minus the cost of 2x, minus 5x times the cost of 2x divided by 2, plus 5 times the sine of 2x divided by 4, plus our integration constant c...