distinct quadratic factors∫(10dx)/(4x^(3)-4x^(2)+5x)

Question

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distinct quadratic factors of ∫(10dx)/(4x^(3)-4x^(2)+5x)

Answer 1

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).