00:01
For this problem, we are to evaluate the definite integral from 0 to 1 of 1 plus x plus x cubed over the square of 1 plus x squared using partial fraction decomposition.
00:11
Now 1 plus x cubed over the square of 1 plus x squared.
00:16
This is equal to the sum of partial fractions whose denominators are 1 plus x squared and we have 1 plus x squared.
00:27
So since they're both quadratic, then their numerators must be linear.
00:33
Let's call them ax plus b and cx plus d.
00:38
And then we will find the values of a, b, c, and d.
00:41
We multiply this first by the lcd.
00:44
That's 1 plus x squared squared.
00:46
So we will get 1 plus x plus x cubed.
00:50
That's equal to ax plus b plus cx plus d times 1 plus x squared.
00:58
Squared, we will get 1 plus x plus x cubed, that's equal to a x plus b, plus cx raised to the third power, plus d x squared, plus cx plus d, and then you will combine like terms.
01:18
You get 1 plus x plus x cubed, that's equal to b plus d, plus we have a plus c times x, plus plus d x squared plus cx cubed.
01:35
And then we will compare the coefficients that we have.
01:38
Now b plus d is the constant term, and we will compare it to the constant term at the left -hand side.
01:44
So b plus d equals 1.
01:46
A plus c is the coefficient of x.
01:49
So since the left -hand side is a coefficient of 1 for x, that means a plus c is equal to 1.
01:56
We don't have x squared, so d is equal to 0.
02:01
And for x cubed, since the coefficient at the left hand side is 1, we have c equal to 1.
02:09
So if c is equal to 1 and a plus c equals 1, that means a is equal to 0...