What is the integral of (2x^2-x)/(x^2-x+2) ?

The integral of (2x^2-x)/(x^2-x+2) is 2x+1/2 ln|4x^2-4x+8|-sqrt(7)arctan(1/(sqrt(7))(2x-1))+C

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integral of (2x^2-x)/(x^2-x+2)

\int \frac{2 x ^{2} - x}{x ^{2} - x + 2} d x

2 x + \frac{1}{2} ln 4 x^{2} - 4 x + 8 - 7 arctan (\frac{1}{7} (2 x - 1)) + C

Solution steps

\int \frac{2 x ^{2} - x}{x ^{2} - x + 2} d x

Long division

\frac{2 x ^{2} - x}{x ^{2} - x + 2} : 2 + \frac{x - 4}{x ^{2} - x + 2}

= \int 2 + \frac{x - 4}{x ^{2} - x + 2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 d x + \int \frac{x - 4}{x ^{2} - x + 2} d x

\int 2 d x = 2 x

\int \frac{x - 4}{x ^{2} - x + 2} d x = \frac{1}{2} ln 4 x^{2} - 4 x + 8 - 7 arctan (\frac{1}{7} (2 x - 1))

= 2 x + \frac{1}{2} ln 4 x^{2} - 4 x + 8 - 7 arctan (\frac{1}{7} (2 x - 1))

Add a constant to the solution

= 2 x + \frac{1}{2} ln 4 x^{2} - 4 x + 8 - 7 arctan (\frac{1}{7} (2 x - 1)) + C

What is the integral of (2x^2-x)/(x^2-x+2) ?
The integral of (2x^2-x)/(x^2-x+2) is 2x+1/2 ln|4x^2-4x+8|-sqrt(7)arctan(1/(sqrt(7))(2x-1))+C