Integral-derivative issue The derivative of $sin^2(x)$ is $2sin(x)cos(x)$. You can also write it as $sin(2x)$.

If we integrate $\sin(2x)$ we get $-0.5\cos(2x)$ and according to calculator does not equal $\sin^2(x)$. Help?


Both functions ($\sin^2$ and $-\frac 12 \cos(2\cdot)$) differ by a constant, as any two function $\mathbf R \to \mathbf R$ having the same derivative do. Note that 
\begin{align*}
  \cos(2x) &= \cos^2 x - \sin^2 x\\
           &= 1 - \sin^2 x - \sin^2 x\\
           &= 1 - 2\sin^2 x\\
\iff \sin^2 x &= \frac 12 - \frac 12\cos(2x) 
\end{align*}
