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

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Answer 1 · 2018-04-18T20:41:13

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

$u = 1- x$

$du = -dx$

$dx = -du$

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 $x$

$-ln | 1-x | + 1- x$

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