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

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

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

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

- x + 6 ln ∣ x + 4 ∣ - 4 + C

Solution steps

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

Expand

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

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

\int \frac{2}{x + 4} d x = 2 ln ∣ x + 4 ∣

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

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

Simplify

= - x + 6 ln ∣ x + 4 ∣ - 4

Add a constant to the solution

= - x + 6 ln ∣ x + 4 ∣ - 4 + C

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