Question

How do you find the integral of `(4x)/(4x+7)dx`?

Answer 1

I like rewriting: `(4x)/(4x+7) = (4x+7-7)/(4x+7) = 1- 7/(4x+7)`

Explanation:

`int (4x)/(4x+7) = int( 1- 7/(4x+7)) dx`

`= x-7/4 ln abs(4x+7) +C`

Note
To evaluate `int 7/(4x+7) dx` use substitution with `u= 4x+7` so `dx = 1/4 du` and we have `7/4 int 1/u du`

Second Note

This integral can also be evaluated by integration by parts, with `u=x` and `dv = 4/(4x+7)`.

Parts gives us:

`xln abs(4x+7) - int ln (4x+7) dx`

The integral here can be found by substitution if you know `int lnu du` and by substitution and integration by parts if you don't know `int ln u du`.