Sum $\dfrac{1}{ 8 + 2x^2}$

Question

Here's another one from my gf's calc 2 midterm.
The professor provided an answer key but his work doesn't make sense.
Attached is the work I did to do the problem, and his work from the answer key.
Specifically, his last steps of integration don't quite seem to make sense to me. Any insight would be appreciated. Thank you.enter image description here

Answer 1

But, notice that when using the integral test, $\sum a_n $ is not equal $\int f(n)$ where $a_n = f(n)$. It is just an approximation that only works as long as $f'(x) < 0$ on $[a, \infty)$ for some $a$. Also, $f$ better be positive and continuous. In conclusion, both yours and your professor solution may not be correct