It's hard to tell where you made an error if you don't show your working, your result, or even the problem you're trying to solve.
But that said, I do know a lot of people have trouble seeing "the constants of integration are just different" when it's not directly obvious. For example, they would have trouble recognizing that both solutions
$$ \sin^2 x + C_1 \qquad \qquad -\cos^2 x + C_2 $$
are the same solution, because they did not think to check beyond simply adding a number to the formal expression.
If you really think you have two equal results, consider it a new math problem to see if the two results are the same or different. e.g. do one of the following two things:
Solve for the constant $A$ in $\sin^2 x + A = -\cos^2 x$
to see that they really are equal, or
Find two values $a$ and $b$ so that $(\sin^2 a) - (-\cos^2 a) \neq (\sin^2 b) - (-\cos^2 b)$ have different values.
to see that they really aren't equal. (and so there really must be some mistake or oversight)