Question

Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`

Answer 1

Let I = `int (2x)/(4 - 3x - x^2).dx`

Let `(2x)/(4 - 3x - x^2)`

= `(2x)/((4 + x)(1 - x)`

= `"A"/(4 + x) + "B"/(1 - x)`

∴ 2x = A(1 – x) + B(4 + x)
Put 4 + x = 0, i.e. x = – 4, we get
– 8 = A(5) + B(0)

∴ A = `-(8)/(5)`
Put 1 – x = 0, i.e x = 1, we 
2 = A(0) + B(5)

∴ B = `(2)/(5)`

∴ `(2x)/(4 - 3x - x^2) = ((-8/5))/(4 + x) + ((2/5))/(1 - x)`

∴ I = `int [((-8)/5)/(4 + x) + ((2/5))/(1 - x)].dx`

= `-(8)/(5) int(1)/(4 + x).dx + (2)/(5) int (1)/(1 - x).dx`

= `-(8)/(5)log|4 + x| + (2)/(5).(log|1 - x|)/(-1) + c`

= `-(8)/(5)log|4 + x| - (2)/(5)log|1 - x| + c`.