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

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

Popular >

integral of x/(x+3)

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

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

= \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

\frac{\partial}{\partial y} (x + e^{y})

t \to 3 lim (2 t^{- 1})

\frac{d}{d x} (\frac{5 + 6 x - 8 x}{x})

\frac{d ^{2}}{d x ^{2}} (e^{\frac{2}{x}})

\int \frac{( x ^{2} - x + 28 )}{x ^{3} + 7 x} d x