Question

Integrate the following w.r.t.x: `(x + 5)/(x^3 + 3x^2 - x - 3)`

Answer 1

Let I = `int (x + 5)/(x^3 + 3x^2 - x - 3)*dx`

= `int (x + 5)/(x^2(x + 3) - (x + 3))*dx`

= `int (x + 5)/((x + 3)(x^2 - 1)`

= `int (x + 5)/((x + 3)(x - 1)(x + 1))*dx`

∴ x² + 2 = A(x + 2)(x + 3) + B(x – 1)(x + 3) + C(x – 1)(x + 2)
Put x – 1 = 0, i.e. x = 1, we get
1 + 2 = A(3)(4) + B(0)(4) + C(0)(3)

∴ 3 = 12A

∴ A = `(1)/(4)` 
Put x + 2 = 0, i.e. x = – 2, we get
4 + 2 = A(0)(1) + B(– 3)(1) + C(– 3)(0)
∴ 6 = – 3B
∴ B = – 2
Put x + 3 = 0, i.e. x = – 3we get
9 + 2 = A(– 1)(0) + B(– 4)(0) + C(– 4)(– 1)
∴ 11 = 4C

∴ C = `(11)/(4)`

∴ `(x^2 + 2)/((x - 1)(x + 2)(x + 3)) = ((1/4))/(x - 1) + (-2)/(x + 1) + ((11/4))/(x + 3)`

∴ I = `int [((1/4))/(x - 1) + (-2)/(x + 1) + ((11/4))/(x + 3)].dx`

= `(1)/(4) int (1)/(x - 1).dx - 2 int(1)/(x + 1).dx + (11)/(4) int (1)/(x + 3).dx`

`(3)/(4)log|x - 1| - log|x + 1| + (1)/(4)log|x + 3| + c`.