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

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

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

\int \frac{5 x - 2}{x ^{2} - 2 x} d x

4 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

Solution steps

\int \frac{5 x - 2}{x ^{2} - 2 x} d x

Expand

\frac{5 x - 2}{x ^{2} - 2 x} : \frac{5 x}{x ^{2} - 2 x} - \frac{2}{x ^{2} - 2 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{5 x}{x ^{2} - 2 x} d x - \int \frac{2}{x ^{2} - 2 x} d x

\int \frac{5 x}{x ^{2} - 2 x} d x = 5 ln ∣ x - 2 ∣

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

= 5 ln ∣ x - 2 ∣ - (- ln ∣ x ∣ + ln ∣ x - 2 ∣)

Simplify

= 4 ln ∣ x - 2 ∣ + ln ∣ x ∣

Add a constant to the solution

= 4 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

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