Unable to find my mistake in the process of solving a seemingly simple integration problem
Solution 1:
The assumption that one solution is incorrect is false. Both the solutions are correct.
From the comments to the question given by @lulu and @RobertShore makes it clear that exploiting the fact $\tan^{-1}x+\tan^{-1}\frac{1}{x}$ is a constant, we can conclude both solutions are correct.
Further, it is easy to visualize through the following plot showing both the solutions. First solution is blue, Second solution is Red.
