Yes, if you let then so that .
Also .
But then which is certainly NOT "1/(u+1)".
Yes, I think that was a typo/slip. Just in case a picture helps - in confirming your reasoning re the typo but then also getting past your alleged 1/(u+1)...
... where (key in spoiler) ...
Spoiler:
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!