1) With sqrt(1 + x^2) in the denominator you'll find sinh^-1 an easier sub.

Just in case a picture helps...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

Spoiler:

Ok, your way is equally good, and to answer your question, actually...

ln|(sqrt(9+x^2) / 3) + (x/3)|

= ln|(sqrt(9+x^2 + x) / 3)|

= ln|sqrt(9+x^2 + x)| - ln|3|

and ln|3| can be part of the constant.

Anyway, hope you were not too unpleasantly diverted by the pics.

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

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