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

The integral of (4x+4)/(x+4) is 4x+16-12ln|x+4|+C

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

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

4 x + 16 - 12 ln ∣ x + 4 ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

4 (x + 4 - 4 ln ∣ x + 4 ∣) + 4 ln ∣ x + 4 ∣ : 4 x + 16 - 12 ln ∣ x + 4 ∣

= 4 x + 16 - 12 ln ∣ x + 4 ∣

Add a constant to the solution

= 4 x + 16 - 12 ln ∣ x + 4 ∣ + C

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What is the integral of (4x+4)/(x+4) ?
The integral of (4x+4)/(x+4) is 4x+16-12ln|x+4|+C