Question

Calculus. ∫ln(x+5)dx

I can do substitution and then the next step of integration by parts. My question is where does the "-5" go? (See pic) I even plug it into wolfram alpha and the -5 disappears because of "restricted x values"?


(X+5)ln(x+5)-(x+5)+c
Becomes
(X+5)ln(x+5)-x+c

Answer 1

definition of ln, recall x+5>0, x>-5

Answer 2

domain definition

Answer 3

So the subtract 5 in the polynomial goes away because the domain of natural log must be greater than zero?...

I'm still not sure I understand why these two expressions are equal.

(X+5)ln(x+5) - x - 5 + c
Is equivalent to
(X+5)ln(x+5) - x + c

Or is it because -5 is a constant, so -5 is just absorbed into +c?

Answer 4

I think the subtract 5 gets absorbed by the constant, c, because when the anti derivative has a constant attached at the end it just signifies a family of functions.

Thanks for your help, it got me thinking in a different direction.

Answer 5

yes you are correct