integral (4x^2-3x+2)/(4x^2-4x+3) dx

Answer 1

long division shows that we have

∫ 1 + (x-1)/(4x^2-4x+3)
= ∫ 1 + (x-1)/[(2x-1)^2 + 2] dx

let (2x-1)^2 = 2tan^2θ, so
(2x-1)^2 + 2 = 2tan^2θ+2 = 2sec^2θ

2x-1 = √2 tanθ
x = (√2 tanθ+1)/2
x-1 = (√2 tanθ-1)/2
dx = 1/2 √2 sec^2θ

∫ 1 + (x-1)/[(2x-1)^2 + 2] dx
= ∫ (1 + (√2 tanθ-1)/(4sec^2θ)) 1/√2 sec^2θ dθ
. . .
= 1/8(x + 1/8 log(4x^2-4x+3) - 1/(4√22) arctan (2x-1)/√2