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

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

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

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

x + 2 - 4 ln ∣ x + 2 ∣ + C

Solution steps

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

Expand

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

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

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

= x + 2 - 2 ln ∣ x + 2 ∣ - 2 ln ∣ x + 2 ∣

Simplify

= x + 2 - 4 ln ∣ x + 2 ∣

Add a constant to the solution

= x + 2 - 4 ln ∣ x + 2 ∣ + C

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