Can we solve the recurrence relation
$$a_{n + 1} = a_n\left(1 + \dfrac{1}{n + 1}\right) + \dfrac{1}{n + 1}$$
using integrals or mellin transforms? I have tried this one for quite considerable amount of time and I got various ways to solve the same recurrence using simple techniques and found that $a_n = n$ for all $n \geqslant 1$, however I'm wondering if there's a nice solution to this recurrence using some advanced techniques such as integral transforms or exponential generating function or other advanced techniques. I'm most interested in a solution making use of integrals and mellin transforms, differential equations and stuff.
Your help would be highly appreciated. Thanks in advance.