What is my error in this seemingly simple integration? $ \int \frac {-x}{x+1} dx $

Question

This is what I've got: $$ \int \frac {-x}{x+1} dx =-1 \int \frac {x}{x+1} dx $$ Then I create the following substitutions: $ u=x+1 $ , $ du=dx $, $ x=u-1 $, which gives me: $$ -1 \int \frac {u-1}{u} du $$ I proceed: $ -1 [\int \frac {u}{u} du - \int \frac{1}{u} du] $ = $-1 [u- ln|u|]+C $ = $ -1[x+1-ln|x+1|]+C $

Then Finally: $$ -x-1+ln|x+1|+C $$

My TI89 says the correct answer is $$ ln|x+1|-x $$ . What happened to my -1?

Answer 1

Hint:$$\large-1+C={ C }_{\text{new constant}}$$