I am not mathematician but an engineer. At some stage of derivation for a formulation I've just been stuck with the following integration which might be super easy and stupid question for all of you as a professional mathematician but your help is so much valuable for me.
$$ \delta J = \int \frac{du}{dx} \frac{d (\delta u)}{dx} dx $$
where $$ \delta $$ is variational operator which functions as
$$ \frac{d(\delta u)}{dx} = \delta \frac{du}{dx} $$
and
$$ \int \delta dx = \delta \int u dx $$
for differential and integral operators.
The term $$ \delta J$$ is supposed to yield to
$$ \int \frac{du}{dx} \frac{d (\delta u)}{dx} dx = \delta \int \frac{1}{2} (\frac{du}{dx})^2 dx $$
To be able to get this what I did was to take the variational operator out of the integral as the commutative rules are applied as instructed above.
$$ \int \frac{du}{dx} \delta \frac{du}{dx} dx = \delta \int \frac{du}{dx} \frac{du}{dx} dx = \delta \int (\frac{du}{dx})^2 dx $$
where 1/2 doesn't appear.
As a result I am not able to understand where the 1/2 coefficient is coming from.
