00:01
All right, so today we're going to take the integral of this function from zero to one.
00:05
In order to do so, we're going to first need to do long division because the exponent on top is larger than that on the bottom.
00:13
So we'll go ahead and set that up.
00:26
All right, how many times does x squared go into x cubed? we'll say x, then x times this expression you should get x cubed minus x squared minus 6x.
00:42
And don't forget for division, this whole expression needs to be subtractive.
00:48
You know you have the correct expression up here when the first term cancels after the subtraction.
00:56
So x cubed minus x cubed will be zero.
00:59
Then we'll have a positive x squared.
01:04
And we also have a positive 6x.
01:06
So we have negative 3x plus 6x.
01:11
And then the negative 11 comes down.
01:14
And we repeat.
01:15
How many times does x squared go into xx? squared, that would be one.
01:21
Then do the multiplication.
01:22
You should have x squared minus x minus six.
01:29
That's just one times this expression.
01:32
Again, subtract the whole thing in parentheses.
01:36
The first term x squared cancels.
01:38
Then you have 3x plus x which gives you 4x.
01:43
Negative 11 plus 6 will be minus 5.
01:48
And because the exponent of this expression is lower than the one we're dividing, we know that this is our final answer.
01:55
So if we want to do the integral now, then we will rewrite this as the x plus one.
02:10
And this 4x minus 5 is going to be the remainder, which will still be divided by the denominator.
02:28
All right, we have an integral from 0 to 1.
02:31
Now we can go ahead.
02:32
We'll do the integral for the first two terms.
02:36
That should be 1, half x squared plus x evaluated from zero to one.
02:44
And we're also going to have this integral here.
03:02
Notice that because the bottom bound is zero, we really just need to substitute one.
03:07
So substitute one in for x, and you should have one half plus one.
03:14
And then we have this integral.
03:16
Now in order to do this integral, i noticed that the bottom of the fraction can be factored.
03:23
So let's do that.
03:27
We'll simplify this to three halves right there, the one -half plus one.
03:36
And we're also going to factor the denominator...