4x3-4x2+5x=x(4x2-4x+5)=x(4x2-4x+1+4)=x((2x-1)2+4).
10/(4x3-4x2+5x)=10[A/x+(Bx+C)/(4x2-4x+5)],
1=4Ax2-4Ax+5A+Bx2+Cx.
Matching coefficients:
constant: 5A=1, A=⅕
x: -4A+C=0, C=⅘
x2: 4A+B=0, B=-⅘
Let J=∫10dx/(4x3-4x2+5x)=2∫dx/x+∫(8-8x)dx/[((2x-1)2+4)],
J=2∫dx/x-∫(8x-4+4)dx/[((2x-1)2+4)],
J=2∫dx/x-∫(8x-4)dx/[((2x-1)2+4)]-4∫dx/[((2x-1)2+4)].
Let y=4x2-4x+5, dy=(8x-4)dx, dx=dy/(8x-4).
J=2∫dx/x-∫dy/y-4∫dx/[((2x-1)2+4)].
J=ln(x2)-ln|y|-K, where K=4∫dx/[((2x-1)2+4)].
Let 2x-1=2tanθ, then 2dx=2sec2θdθ, dx=sec2θdθ.
(2x-1)2+4=4tan2θ+4=4sec2θ.
K=4∫sec2θdθ/(4sec2θ)=∫dθ=θ=tan-1[(2x-1)/2].
J=ln(x2)-ln(4x2-4x+5)-tan-1[(2x-1)/2]+C, where C is a constant,
J=ln|ax2/(4x2-4x+5)|-tan-1[(2x-1)/2], where a is a constant=ln(C).