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

The integral of (3x+2)/(x-3) is 3x-9+11ln|x-3|+C

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

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

3 x - 9 + 11 ln ∣ x - 3 ∣ + C

Solution steps

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

Expand

\frac{3 x + 2}{x - 3} : \frac{3 x}{x - 3} + \frac{2}{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{3 x}{x - 3} d x + \int \frac{2}{x - 3} d x

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

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

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

Simplify

3 (x - 3 + 3 ln ∣ x - 3 ∣) + 2 ln ∣ x - 3 ∣ : 3 x - 9 + 11 ln ∣ x - 3 ∣

= 3 x - 9 + 11 ln ∣ x - 3 ∣

Add a constant to the solution

= 3 x - 9 + 11 ln ∣ x - 3 ∣ + C

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What is the integral of (3x+2)/(x-3) ?
The integral of (3x+2)/(x-3) is 3x-9+11ln|x-3|+C