Question

How do you use substitution to integrate `x/(1-x)`?

Answer 1

`int ( x/(1-x) ) dx`

`u = 1- x`

`du = -dx`

`dx = -du`

Substitute back in

`int ( x/u )* -du`

We still have an `x` in the problem, so let's use our `u` substitution to solve for `x`:

`u = 1-x`

`x = 1-u`

Now we have

`-int ( (1-u)/u )* du`

`- int ( 1/u - u/u )`

Split it up

`- int ( (1)/u )* du + int (u/u) du`

`- int ( (1)/u )* du + int du`

`- ln|u| + u`

Get back in terms of `x`

`-ln | 1-x | + 1- x`