Question

How do you use partial fraction decomposition to decompose the fraction to integrate `(13x) / (6x^2 + 5x - 6)`?

Answer 1

Using a fraction whose denominator is quadratic, you need to check whether or not it is factorable.

`6x^2 + 5x - 6 = (2x+3)(3x-2)`

Since it is, we only need to write out the following:

`int (13x)/((2x + 3)(3x - 2))dx = int A/(2x + 3) + B/(3x - 2)dx`

To find what `A` and `B` are, you could start by cross-multiplying. For now, ignore the integral sign and `dx`.

`= (A(3x-2) + B(2x+3))/((2x+3)(3x - 2))`

`= (3Ax - 2A + 2Bx+3B)/((2x+3)(3x - 2))`

`= ((3A + 2B)x + (-2A + 3B))/((2x+3)(3x - 2))`

Thus, we have the equations:

`-2A + 3B = 0 => B = 2/3 A`

`3A + 2B = 13 => 3A + 4/3 A = 13`
`13/3 A = 13`
`:. color(green)(A = 3) => color(green)(B = 2)`

At this point we are pretty much done.

`= int 3/(2x + 3) + 2/(3x - 2)dx`

`= color(blue)(3/2ln|2x + 3| + 2/3ln|3x - 2| + C)`