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Question
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Solution
Let I = `int (12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)).dx`
I = `int ((12x^2 - 3) - 2x - 6)/((4x^2 - 1).(x + 3))dx`
I = `int (3(4x^2 - 1) - 2 (x + 3))/((4x^2 - 1).(x + 3))dx`
I = `int (3(4x^2 - 1))/((4x^2 - 1).(x + 3))dx - int (2(x + 3))/((4x^2 - 1).(x + 3)) dx`
I = `3.int 1/(x + 3) dx - 2 int 1/((4x^2 - 1))dx`
I = `3.log |x + 3| - (1/2)/(4/2) int 1/ (x^2 - 1/4)dx + c_1`
I = `3.log |x + 3| - 1/2 int 1/ (x^2 - (1/2)^2)dx + c_1`
I = `3.log |x + 3| - 1/2 xx 1/(2(1/2)). log |x - 1/2|/|x + 1/2| + c_1 + c_2`
I = `3. log |x + 3| - 1/2 log |2x - 1|/|2x + 1| + c`
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