Is there anything that we can say about $f$ in terms of $\theta$ if $\int_0^tf(\tau)\text{d}\tau = \theta(t)\int_0^t\theta(\tau)\text{d}\tau$ is true?

Question

I have an integral equation in which I have $\theta(t)\int_0^t\theta(\tau)\text{d}\tau$ as a term, which I would like to transform to a single integral, so that I end up with a Volterra integral equation. Basically I'm looking for an $f$ function for which $$\int_0^tf(\tau)\text{d}\tau = \theta(t)\int_0^t\theta(\tau)\text{d}\tau$$ is true. Is there a way to do this in general? If not, is there a family of functions that this can be done?

We can assume $\theta$ to be smooth as it is a pendulum's angle in time.

Answer 1

Differentiating both sides, $$ f(t) = \theta'(t) \int_0^t \theta(\tau)\; d\tau + \theta(t)^2$$