Why do you need absolute value signs when an integral of something is a natural log?

Well think about it you can't take the Ln of any negative number unless your working with complex numbers. So the Absolute value is just there to turn the negative into positive.

Example Ln|-10| would be correct
But Ln(-10) would be a non-existent solution

The ln function is only defined when its argument is positive. (In your case the argument is x+5). And to ensure that an argument is positive, we sometimes need to put absolute value signs around it.

Eg: x+5 can be negative (when x<5), so we need ln|x+5| not ln(x+5)
      But x^2 +1 is always positive, so ln(x^2+1) is fine

Also, ln is not defined when its argument is zero. So I suppose technically for your integral your answer should be "ln|x+5| + constant, where x cannot equal -5", however, simply "ln|x+5| + constant" is generally written.