Let $V\subset H\subset V^*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^*),$$ where $f\in\ L^2(0,T;H)$.

Can we let $f\in L^2(0,T;V^*)$?

This question is a special case ($A(t)=A$) of Regularity of solution to a hyperbolic pde. There, the answer says

If you want $f$ to take values in $V^*$ rather than $H$, you can do this if you assume more temporal regularity on $f$. Basically, the idea is to integrate by parts in the term $\int_0^t <u',f>$ in the energy estimate. You will have no trouble finding results of this type in the literature.

I think by integration by parts it is meant $$\int_0^t <f(s),v'(s)>_{V^*,V}ds=(f(t),v(t))_H-(f(0),v(0))_H-\int_0^t <v(s),f'(s)>_{V^*,V} ds,$$ but the right hand side does not make sense unless $f'(s)\in V$, and I am confused. I am actually having trouble finding results of this type in the literature.

Where can I find this type of result?