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Question
Integrate the following with respect to x.
`(3x + 2)/((x - 2)(x - 3))`
Solution
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"`
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