Question

Integrate the following with respect to x.

`(3x + 2)/((x - 2)(x - 3))`

Answer 1

We use partial fraction method to split the given function into two fractions and then integrate.

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

3x + 2 = A(x − 3) + B(x − 2)

Put (x = 3)

11 = B

⇒ B = 11

Put (x = 2)in the identify

39) + 2 = A(2 - 3)

8 = − A

⇒ A = − 8

So `(3x + 2)/((x - 2)(x - 3)) = (-8)/(x - 2) + 11/(x - 3)`

Thus `int (3x + 2)/((x - 2)(x - 3))   "d"x = 8 int ("d"x)/(x - 2) + 11 int ("d"x)/(x - 3) + "c"`

= `- 8 log |x - 2| + 11 log |x - 3| + "c"`

`int (3x + 2)/((x - 2)(x - 3))  "d"x   11 log |x - 3| - 8 log |x - 2| + "c"`