What is the integral of (2x^2-x+1)/(2x^2-3x+1) ?

The integral of (2x^2-x+1)/(2x^2-3x+1) is x-ln|2x-1|+2ln|x-1|+C

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integral of (2x^2-x+1)/(2x^2-3x+1)

\int \frac{2 x ^{2} - x + 1}{2 x ^{2} - 3 x + 1} d x

x - ln ∣ 2 x - 1 ∣ + 2 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{2 x ^{2} - x + 1}{2 x ^{2} - 3 x + 1} d x

Take the partial fraction of

\frac{2 x ^{2} - x + 1}{2 x ^{2} - 3 x + 1} : 1 - \frac{2}{2 x - 1} + \frac{2}{x - 1}

= \int 1 - \frac{2}{2 x - 1} + \frac{2}{x - 1} 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 1 d x - \int \frac{2}{2 x - 1} d x + \int \frac{2}{x - 1} d x

\int 1 d x = x

\int \frac{2}{2 x - 1} d x = ln ∣ 2 x - 1 ∣

\int \frac{2}{x - 1} d x = 2 ln ∣ x - 1 ∣

= x - ln ∣ 2 x - 1 ∣ + 2 ln ∣ x - 1 ∣

Add a constant to the solution

= x - ln ∣ 2 x - 1 ∣ + 2 ln ∣ x - 1 ∣ + C

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What is the integral of (2x^2-x+1)/(2x^2-3x+1) ?
The integral of (2x^2-x+1)/(2x^2-3x+1) is x-ln|2x-1|+2ln|x-1|+C