What is the integral of (2x)/(x+2) ?

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

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integral of (2x)/(x+2)

\int \frac{2 x}{x + 2} d x

2 (x + 2 - 2 ln ∣ x + 2 ∣) + C

Solution steps

\int \frac{2 x}{x + 2} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x + 2} d x

Apply u-substitution

: \int \frac{u - 2}{u} d u

= 2 \cdot \int \frac{u - 2}{u} d u

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

= 2 \cdot \int 1 - \frac{2}{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

= 2 (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 2 (u - 2 ln ∣ u ∣)

Substitute back

u = x + 2

= 2 (x + 2 - 2 ln ∣ x + 2 ∣)

Add a constant to the solution

= 2 (x + 2 - 2 ln ∣ x + 2 ∣) + C

\frac{d y}{d x} = 0.0225 y + 0.0003 y^{2}

\frac{d}{d x} (\frac{2}{( x - 2 ) ^{3}})

\frac{\partial}{\partial x} (x^{2} y + 3 x y + y^{2})

x \to \infty lim ((\frac{( 2 x - 3 )}{( 2 x + 5 )})^{2 x + 1})

x = \frac{4}{y}, x = \frac{y}{8}, x = 8 y

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