What is the integral of ((2x-6x+4))/((x-3)) ?

The integral of ((2x-6x+4))/((x-3)) is -4x+12-8ln|x-3|+C

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integral of ((2x-6x+4))/((x-3))

\int \frac{( 2 x - 6 x + 4 )}{( x - 3 )} d x

- 4 x + 12 - 8 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{2 x - 6 x + 4}{x - 3} d x

Expand

\frac{2 x - 6 x + 4}{x - 3} : \frac{2 x}{x - 3} - \frac{6 x}{x - 3} + \frac{4}{x - 3}

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 - 3} d x - \int \frac{6 x}{x - 3} d x + \int \frac{4}{x - 3} d x

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

\int \frac{6 x}{x - 3} d x = 6 (x - 3 + 3 ln ∣ x - 3 ∣)

\int \frac{4}{x - 3} d x = 4 ln ∣ x - 3 ∣

= 2 (x - 3 + 3 ln ∣ x - 3 ∣) - 6 (x - 3 + 3 ln ∣ x - 3 ∣) + 4 ln ∣ x - 3 ∣

Simplify

2 (x - 3 + 3 ln ∣ x - 3 ∣) - 6 (x - 3 + 3 ln ∣ x - 3 ∣) + 4 ln ∣ x - 3 ∣ : - 4 x + 12 - 8 ln ∣ x - 3 ∣

= - 4 x + 12 - 8 ln ∣ x - 3 ∣

Add a constant to the solution

= - 4 x + 12 - 8 ln ∣ x - 3 ∣ + C

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What is the integral of ((2x-6x+4))/((x-3)) ?
The integral of ((2x-6x+4))/((x-3)) is -4x+12-8ln|x-3|+C