indefinite integration

For the integration of v/(4+3v) dv

we first need to express the numerator in terms of denominator so as

to simplify the expression, so that it can be integrated using simpler

formulas.

Now, 

v/4+3v = 3v/3(4+3v)

 = 3v+4-4/3(4+3v)

 = 1/3  - 4/3(4+3v)

 

So the integral becomes,

?1/3 dv - 4/3?dv/(4+3v)

= v/3  - 4/9 ln(4+3v)

 

This is the answer