Integral of (x-4)/(x^2-x+2)

1. The problem statement, all variables and given/known data

[itex]\frac{x-4}{x^2-x+2}[/itex]





2. Relevant equations







3. The attempt at a solution



[itex]\frac{x-4}{(x-1/2)^2+7/4}[/itex]

u = x - 1/2 = [itex]\frac{\sqrt{7}}{2}[/itex]tan[itex]\theta[/itex] ; du = [itex]\sqrt{7}[/itex]/2 sec^2(theta)

x = u + 1/2

[itex]\int\frac{(u−7/2)(\sqrt{7}/2sec^2(\theta)}{7/4 sec^2(\theta)}[/itex]



This simplifies to: [itex]\int tan(\theta) - \sqrt{7}[/itex]

= ln(sec(theta)) - [itex]\sqrt{7}[/itex](theta)

= ln([itex]\frac{\sqrt{(2x-1)^2+7}}{\sqrt{7}}[/itex]) - [itex]\sqrt{7}[/itex]arctan([itex]\frac{2x-1}{\sqrt{7}}[/itex]) + C

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