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

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

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integral of (2u)/(u+2)

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

2 (u + 2 - 2 ln ∣ u + 2 ∣) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Expand

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

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

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 v - \int \frac{2}{v} d v)

\int 1 d v = v

\int \frac{2}{v} d v = 2 ln ∣ v ∣

= 2 (v - 2 ln ∣ v ∣)

Substitute back

v = u + 2

= 2 (u + 2 - 2 ln ∣ u + 2 ∣)

Add a constant to the solution

= 2 (u + 2 - 2 ln ∣ u + 2 ∣) + C

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