What is the integral of x/(x+13) ?

The integral of x/(x+13) is x+13-13ln|x+13|+C

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integral of x/(x+13)

\int \frac{x}{x + 13} d x

x + 13 - 13 ln ∣ x + 13 ∣ + C

Solution steps

\int \frac{x}{x + 13} d x

Apply u-substitution

: \int \frac{u - 13}{u} d u

= \int \frac{u - 13}{u} d u

Expand

\frac{u - 13}{u} : 1 - \frac{13}{u}

= \int 1 - \frac{13}{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{13}{u} d u

\int 1 d u = u

\int \frac{13}{u} d u = 13 ln ∣ u ∣

= u - 13 ln ∣ u ∣

Substitute back

u = x + 13

= x + 13 - 13 ln ∣ x + 13 ∣

Add a constant to the solution

= x + 13 - 13 ln ∣ x + 13 ∣ + C

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What is the integral of x/(x+13) ?
The integral of x/(x+13) is x+13-13ln|x+13|+C