How do you integrate $x/(x+10)$ ?

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Answer 1 · 2018-04-09T18:46:36

$intx/(x+10)dx=x-10ln|x+10|+C$

A simple substitution will do.

$u=x+10 -> x=u-10$

$du=dx$

Rewrite and simplify:

$int(u-10)/udu=intu/udu-10int(du)/u$

Integrate:

$intdu-10int(du)/u=u-10ln|u|+C$

Rewrite in terms of $x,$ yielding

$intx/(x+10)dx=x+10-10ln|x+10|+C$

We may absorb the $10$ into $C.$

$intx/(x+10)dx=x-10ln|x+10|+C$

How do you integrate x/(x+10)x/(x+10)?