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

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

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

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

2 (2 + x - 2 ln ∣ 2 + x ∣) + C

Solution steps

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

= 2 (2 + x - 2 ln ∣ 2 + x ∣)

Add a constant to the solution

= 2 (2 + x - 2 ln ∣ 2 + x ∣) + C

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