What is the integral of x/(x+5) ?

The integral of x/(x+5) is x+5-5ln|x+5|+C

integral of x/(x+5)

\int \frac{x}{x + 5} d x

x + 5 - 5 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x}{x + 5} d x

Apply u-substitution

= \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \int 1 - \frac{5}{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{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u - 5 ln ∣ u ∣

Substitute back

u = x + 5

= x + 5 - 5 ln ∣ x + 5 ∣

Add a constant to the solution

= x + 5 - 5 ln ∣ x + 5 ∣ + C

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