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

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

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

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

- ln ∣ x + 3 ∣ + x + C

Solution steps

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

\frac{x + 2}{x + 3} = - \frac{1}{x + 3} + 1

= \int - \frac{1}{x + 3} + 1 d x

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{1}{x + 3} d x + \int 1 d x

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

\int 1 d x = x

= - ln ∣ x + 3 ∣ + x

Add a constant to the solution

= - ln ∣ x + 3 ∣ + x + C

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