What is the integral of x/((x+3)) ?

The integral of x/((x+3)) is x+3-3ln|x+3|+C

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integral of x/((x+3))

integral

\frac{x}{( x + 3 )}

x + 3 - 3 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{x}{x + 3} d x

Apply u-substitution

= \int \frac{u - 3}{u} d u

Expand

\frac{u - 3}{u} : 1 - \frac{3}{u}

= \int 1 - \frac{3}{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{3}{u} d u

\int 1 d u = u

\int \frac{3}{u} d u = 3 ln ∣ u ∣

= u - 3 ln ∣ u ∣

Substitute back

u = x + 3

= x + 3 - 3 ln ∣ x + 3 ∣

Add a constant to the solution

= x + 3 - 3 ln ∣ x + 3 ∣ + C