What is the integral of (v+1)/(2v+1) ?

The integral of (v+1)/(2v+1) is 1/2 v+1/4+1/4 ln|2v+1|+C

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integral of (v+1)/(2v+1)

\int \frac{v + 1}{2 v + 1} d v

\frac{1}{2} v + \frac{1}{4} + \frac{1}{4} ln ∣ 2 v + 1 ∣ + C

Solution steps

\int \frac{v + 1}{2 v + 1} d v

Expand

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{v}{2 v + 1} d v + \int \frac{1}{2 v + 1} d v

\int \frac{v}{2 v + 1} d v = \frac{1}{4} (2 v + 1 - ln ∣ 2 v + 1 ∣)

\int \frac{1}{2 v + 1} d v = \frac{1}{2} ln ∣ 2 v + 1 ∣

= \frac{1}{4} (2 v + 1 - ln ∣ 2 v + 1 ∣) + \frac{1}{2} ln ∣ 2 v + 1 ∣

Simplify

\frac{1}{4} (2 v + 1 - ln ∣ 2 v + 1 ∣) + \frac{1}{2} ln ∣ 2 v + 1 ∣ : \frac{1}{2} v + \frac{1}{4} + \frac{1}{4} ln ∣ 2 v + 1 ∣

= \frac{1}{2} v + \frac{1}{4} + \frac{1}{4} ln ∣ 2 v + 1 ∣

Add a constant to the solution

= \frac{1}{2} v + \frac{1}{4} + \frac{1}{4} ln ∣ 2 v + 1 ∣ + C

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What is the integral of (v+1)/(2v+1) ?
The integral of (v+1)/(2v+1) is 1/2 v+1/4+1/4 ln|2v+1|+C