What is the integral of (2x-4)/(x(x-3)(x-4)) ?

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

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integral of (2x-4)/(x(x-3)(x-4))

\int \frac{2 x - 4}{x ( x - 3 ) ( x - 4 )} d x

- \frac{2}{3} ln ∣ x - 3 ∣ - \frac{1}{3} ln ∣ x ∣ + ln ∣ x - 4 ∣ + C

Solution steps

\int \frac{2 x - 4}{x ( x - 3 ) ( x - 4 )} d x

Expand

\frac{2 x - 4}{x ( x - 3 ) ( x - 4 )} : \frac{2 x}{x ( x - 3 ) ( x - 4 )} - \frac{4}{x ( x - 3 ) ( x - 4 )}

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}{x ( x - 3 ) ( x - 4 )} d x - \int \frac{4}{x ( x - 3 ) ( x - 4 )} d x

\int \frac{2 x}{x ( x - 3 ) ( x - 4 )} d x = 2 (- ln ∣ x - 3 ∣ + ln ∣ x - 4 ∣)

\int \frac{4}{x ( x - 3 ) ( x - 4 )} d x = ln ∣ x - 4 ∣ - \frac{4}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣

= 2 (- ln ∣ x - 3 ∣ + ln ∣ x - 4 ∣) - (ln ∣ x - 4 ∣ - \frac{4}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣)

Simplify

2 (- ln ∣ x - 3 ∣ + ln ∣ x - 4 ∣) - (ln ∣ x - 4 ∣ - \frac{4}{3} ln ∣ x - 3 ∣ + \frac{1}{3} ln ∣ x ∣) : - \frac{2}{3} ln ∣ x - 3 ∣ - \frac{1}{3} ln ∣ x ∣ + ln ∣ x - 4 ∣

= - \frac{2}{3} ln ∣ x - 3 ∣ - \frac{1}{3} ln ∣ x ∣ + ln ∣ x - 4 ∣

Add a constant to the solution

= - \frac{2}{3} ln ∣ x - 3 ∣ - \frac{1}{3} ln ∣ x ∣ + ln ∣ x - 4 ∣ + C

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