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

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

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integral of x/(x+4)

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

x + 4 - 4 ln ∣ x + 4 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 4}{u} d u

= \int \frac{u - 4}{u} d u

Expand

\frac{u - 4}{u} : 1 - \frac{4}{u}

= \int 1 - \frac{4}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int \frac{4}{u} d u

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= u - 4 ln ∣ u ∣

Substitute back

u = x + 4

= x + 4 - 4 ln ∣ x + 4 ∣

Add a constant to the solution

= x + 4 - 4 ln ∣ x + 4 ∣ + C

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