What is the integral of (6x+5)/(x+2) ?

The integral of (6x+5)/(x+2) is 6x+12-7ln|x+2|+C

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integral of (6x+5)/(x+2)

\int \frac{6 x + 5}{x + 2} d x

6 x + 12 - 7 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{6 x + 5}{x + 2} d x

Expand

\frac{6 x + 5}{x + 2} : \frac{6 x}{x + 2} + \frac{5}{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{6 x}{x + 2} d x + \int \frac{5}{x + 2} d x

\int \frac{6 x}{x + 2} d x = 6 (x + 2 - 2 ln ∣ x + 2 ∣)

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

= 6 (x + 2 - 2 ln ∣ x + 2 ∣) + 5 ln ∣ x + 2 ∣

Simplify

6 (x + 2 - 2 ln ∣ x + 2 ∣) + 5 ln ∣ x + 2 ∣ : 6 x + 12 - 7 ln ∣ x + 2 ∣

= 6 x + 12 - 7 ln ∣ x + 2 ∣

Add a constant to the solution

= 6 x + 12 - 7 ln ∣ x + 2 ∣ + C

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What is the integral of (6x+5)/(x+2) ?
The integral of (6x+5)/(x+2) is 6x+12-7ln|x+2|+C