<s>[INST] How to reconcile the fact that the antiderivative of $\sin(x)\cos(x)$ has two possible answers? So here is the problem. 
$$
\int\sin(x)\cos(x)dx = \frac{1}{2}\sin^2(x) + c_1 \\
= -\frac{1}{2}\cos^2(x) + c_2
$$
This fact doesn't make much sense to me. How do you reconcile the fact that there are two possibilities (notice that these answers aren't only different by a constant)? 
What are some other functions that have more than one antiderivatives (not only different by a constant)? 
 [/INST] Since $\ \sin^2(x)+\cos^2(x)=1$, if we let $c_1=c_2-\frac 12$ so we'd have
$$\begin{align}
\frac{1}{2}\sin^2(x) + c_1 &= \frac{1}{2}\sin^2(x) + (c_2-\frac 12) \\
&= \frac{1}{2}\sin^2(x) + (c_2-\frac 12) \\
&= \frac{1}{2}(1-\cos^2(x)) + (c_2-\frac 12) \\
&= c_2-\frac{1}{2}\cos^2(x)
\end{align}$$
</s>