What is the integral of x/(3x+6) ?

The integral of x/(3x+6) is 1/3 x-2/3 ln|3x+6|+C

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integral of x/(3x+6)

\int \frac{x}{3 x + 6} d x

\frac{1}{3} x - \frac{2}{3} ln ∣ 3 x + 6 ∣ + C

Solution steps

\int \frac{x}{3 x + 6} d x

Long division

\frac{x}{3 x + 6} : \frac{1}{3} - \frac{2}{3 x + 6}

= \int \frac{1}{3} - \frac{2}{3 x + 6} d x

Apply the Sum Rule:

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

= \int \frac{1}{3} d x - \int \frac{2}{3 x + 6} d x

\int \frac{1}{3} d x = \frac{1}{3} x

\int \frac{2}{3 x + 6} d x = \frac{2}{3} ln ∣ 3 x + 6 ∣

= \frac{1}{3} x - \frac{2}{3} ln ∣ 3 x + 6 ∣

Add a constant to the solution

= \frac{1}{3} x - \frac{2}{3} ln ∣ 3 x + 6 ∣ + C

sin (2 t) \cdot cos (2 t)

y^{^{''}} + y^{^{'}} = 2

f (x) = 2 x + 1, at x = 4

\frac{d}{d t} (\frac{y}{t} - (\frac{y}{t})^{2})

\int_{1}^{2} \frac{4}{x} - \frac{16}{x ^{2}} d x

What is the integral of x/(3x+6) ?
The integral of x/(3x+6) is 1/3 x-2/3 ln|3x+6|+C