C4 integration!!

Integrating using trig identities.

I came across this example in my book...

$\int sin^2x dx$ which is replaced by $\int \frac{1}{2}(1-cos2x) dx$
the result is then $\frac{1}{2}(x-\frac{1}{2}sin2x) + C$
I don't understand why it is only the bit in the brackets from $\int \frac{1}{2}(1-cos2x) dx$ that gets integrated?
Is there a step inbetween where $\frac{1}{2}$ is moved before the integral sign? e.g. $\frac{1}{2} \int (1-cos2x) dx$ ?

Any help is much appreciated!!