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

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

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

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

4 ln ∣ x - 2 ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

\frac{5 x - 2}{x ( x - 2 )} : \frac{5 x}{x ( x - 2 )} - \frac{2}{x ( x - 2 )}

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 ( x - 2 )} d x - \int \frac{2}{x ( x - 2 )} d x

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

\int \frac{2}{x ( x - 2 )} 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

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