Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

Question

I was working on a list of common integrals and differentials and I came across this question.

If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$

Then why is $$\int \sin(2\theta) \space d\theta = -\frac12\cos(2\theta) + c$$

Isn't integration the opposite of differentiation?

Answer 1

$-\dfrac12 \cos (2\theta)$ is equal to $\sin^2\theta$ up to an arbitrary constant.

Indeed, $-\dfrac12 \cos (2\theta)=-\dfrac12(1-2\sin^2\theta)=\sin^2\theta-\dfrac12$.

Thus, they differ by only $\dfrac12$.

Answer 2

The question is whether $$ \sin^2\theta = -\frac 1 2 \cos(2\theta) + C. $$

A standard trigonometric identity, a half-angle formula, says this is true if $C=\dfrac 1 2$.