[Math] first variation of function defined by an integral

Question

Let $f$ be a function defined by $f(x) = \int_0^x \sin \phi(t) dt$. What is the first variation $\delta f(x)$ and how it is calculated?

Answer 1

There's a wikipedia page First variation with the definition and worked-out example.

In your example, instead of a functional (which would take values in $\mathbb R$) we have a nonlinear operator, which takes values in some function space. But the calculation is the same: $$ \sin(\phi(t)+\epsilon h(t)) - \sin (\phi(t)) = \epsilon \cos(\phi(t))\,h(t)+O(\epsilon^2)$$ where $O(\epsilon^2)$ is uniform in $t$ provided that $h$ is bounded on $[0,x]$. Hence, $$\delta f(x) = \int_0^x \cos(\phi(t))\,h(t)\,dt $$