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Question
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Solution
Let I = `int (3x - 2)/((x + 1)^2(x + 3))*dx`
Let `(3x - 2)/((x + 1)^2(x + 3)) = "A"/(x + 1) + "B"/(x + 1)^2 + "C"/(x + 3)`
∴ 3x – 2 = A(x + 1)(x + 3) + B(x + 3) + C(x + 1)2
Put x + 1 = 0, i.e. x = – 1, we get
– 3 – 2 = A(0)(2) + B(2) + C(0)
∴ – 5 = 2B
∴ B = `-(5)/(2)`
Put x + 3 = 0, i.e. x - – 3, we get
– 9 – 2 = A(– 2)(0) + B(0) + C(– 2)2
∴ – 11 = 4C
∴ C = `-(11)/(4)`
Put x = 0, we get
– 2 = A(1)(3) + B(3) + C(1)
∴ – 2 = 3A + 3B + C
∴ – 2 = `3"A" - (15)/(2) - (11)/(4)`
∴ 3A = `-2 + (15)/(2) + (11)/(4)`
= `(-8 + 30 + 11)/(4)`
∴ A = `(11)/(4)`
∴ `(3x - 2)/((x + 1)^2(x + 3)) = ((11/4))/(x + 1) + ((-5/4))/(x + 1)^2 + ((-11/4))/(x + 3)`
∴ I = `int [((11/4))/(x + 1) + (((-5)/2))/(x + 1)^2 + (((-11)/4))/(x + 3)]`
= `(11)/(4) int 1/(x + 1)*dx - (5)/(2) int (x + 1)^-2*dx - (11)/(4) int 1/(x + 3)*dx`
= `(11)/(4)log|x + 1| -(5)/(2)*(x + 1)^-1/(-1)*(1)/(1)- (11)/(4)log|x + 3| + c`
= `(11)/(4)log|(x + 1)/(x + 3)| + (5)/(2(x + 1)) + c`.
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