After all that excelent math, you have a silly little miscopying!
Yes, integrating gives \(\displaystyle \frac{1}{y}= \frac{1}{36(9x^2+ 2)^2}+ C\).
When x= 0, y= 2 so \(\displaystyle \frac{1}{2}= \frac{1}{36(9(0^2)+ 2)^2}+ C\).
But then you have \(\displaystyle \frac{1}{2}= \frac{1}{36(0+ 1)^2}+ C\) while you should have \(\displaystyle \frac{1}{2}= \frac{1}{36(2)^2}+ C\).
\(\displaystyle \frac{1}{2}= \frac{1}{36(4)}+ C= \frac{1}{144}+ C\).
\(\displaystyle C= \frac{1}{2}- \frac{1}{144}= \frac{72}{144}- \frac{1}{144}= \frac{71}{144}\).