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

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

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integral of v/(v+2)

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

v + 2 - 2 ln ∣ v + 2 ∣ + C

Solution steps

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

Apply u-substitution

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

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

Expand

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

= \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

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= u - 2 ln ∣ u ∣

Substitute back

u = v + 2

= v + 2 - 2 ln ∣ v + 2 ∣

Add a constant to the solution

= v + 2 - 2 ln ∣ v + 2 ∣ + C

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