(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine

[tex]\int\frac{4x^{2}-2x+2}{4x^{2}-4x+3}dx[/tex]

2. Relevant equations

I believe it's necessary to complete the square.

3. The attempt at a solution

Completing the square for

[tex]4x^{2}-4x+3[/tex]

gives

[tex]\int\frac{4x^{2}-2x+2}{(x-\frac{1}{2})^2+\frac{1}{2}}dx[/tex]

let

[tex]u=x-\frac{1}{2}, du = dx, x=u+\frac{1}{2}[/tex]

then

[tex]\int\frac{4(u+\frac{1}{2})^{2}-2(u+\frac{1}{2})+2}{u^2+\frac{1}{2}}du[/tex]

[tex]=\int\frac{4u^2+4u+1-2u-1+2}{u^2+\frac{1}{2}}du[/tex]

[tex]=\int\frac{4u^2+2u+2}{u^2+\frac{1}{2}}du[/tex]

[tex]=\int\frac{4(u^2+\frac{1}{2})}{u^2+\frac{1}{2}}+\int\frac{2u}{u^2+\frac{1}{2}}du[/tex]

[tex]=4u + \ln|u^2+\frac{1}{2}|+c[/tex]

Substituting back

[tex]\int\frac{4x^{2}-2x+2}{4x^{2}-4x+3}dx=4(x-\frac{1}{2})+\ln|(x-\frac{1}{2})^2+\frac{1}{2})|+c[/tex]

[tex]=4(x-\frac{1}{2})+\ln|4x^{2}-4x+3|+c[/tex]

Would someone be so kind as to tell me if this is correct? For this question, I have 4 possible answers (and one that states "None of the above") and my answer doesn't match any of the other 3, so I'm wondering.

Thanks!

phyz

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# Homework Help: Integral of a fraction consisting of two quadratic equations

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