What is the integral of (5x-4)/(x(x^2-2x+2)) ?

The integral of (5x-4)/(x(x^2-2x+2)) is -2ln|x|+ln|x^2-2x+2|+3arctan(x-1)+C

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integral of (5x-4)/(x(x^2-2x+2))

\int \frac{5 x - 4}{x ( x ^{2} - 2 x + 2 )} d x

- 2 ln ∣ x ∣ + ln x^{2} - 2 x + 2 + 3 arctan (x - 1) + C

Solution steps

\int \frac{5 x - 4}{x ( x ^{2} - 2 x + 2 )} d x

Take the partial fraction of

\frac{5 x - 4}{x ( x ^{2} - 2 x + 2 )} : - \frac{2}{x} + \frac{2 x + 1}{x ^{2} - 2 x + 2}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{2 x + 1}{x ^{2} - 2 x + 2} d x = ln x^{2} - 2 x + 2 + 3 arctan (x - 1)

= - 2 ln ∣ x ∣ + ln x^{2} - 2 x + 2 + 3 arctan (x - 1)

Add a constant to the solution

= - 2 ln ∣ x ∣ + ln x^{2} - 2 x + 2 + 3 arctan (x - 1) + C

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What is the integral of (5x-4)/(x(x^2-2x+2)) ?
The integral of (5x-4)/(x(x^2-2x+2)) is -2ln|x|+ln|x^2-2x+2|+3arctan(x-1)+C