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Popular Calculus >

integral of v/(v+1)

Solution

\int \frac{v}{v + 1} d v

Solution

v + 1 - ln ∣ v + 1 ∣ + C

Solution steps

\int \frac{v}{v + 1} d v

Apply u-substitution

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

Expand

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= u - ln ∣ u ∣

Substitute back

u = v + 1

= v + 1 - ln ∣ v + 1 ∣

Add a constant to the solution

= v + 1 - ln ∣ v + 1 ∣ + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of v/(v+1) ?
The integral of v/(v+1) is v+1-ln|v+1|+C