00:01
For this problem, we are to evaluate this indefinite integral using partial fraction decomposition.
00:05
Now, x cubed over x -3 to the 4th power plus 58 x squared plus 4 .1 is equal to x cubed over the product of x squared plus 49 and x squared plus 9.
00:23
So we can write this as a sum of partial fractions with denominators x squared plus 49.
00:30
And the other one is x squared plus 9.
00:33
And since they're both quadratic, then we can use numerators that are linear.
00:38
Let's call them ax plus b and cx plus d.
00:43
Now we want to solve for the values of the coefficients, a, b, c, and d.
00:48
And what we're going to do is multiply this by the lcd, x squared plus 49 times x squared plus 9.
00:55
We then get x cubed equal to a times x plus b times x squared plus 9 plus c times x squared plus d times x squared plus 49.
01:09
We then have x cubed equal to a x raised to the third power plus bx squared plus 9 a x plus 9b plus cx cubed plus 6 cubed plus plus d x squared plus 49c times x plus 49d.
01:37
Combining like terms, we have x cubed equal to a plus c times x cubed plus.
01:46
We have b plus d times x squared plus...