Proving an identity Given $a,b\in\mathbb{R}$ with $a < b$ and defining $F(z):=\int_0^z f(s) \, ds$ with $z \in \mathbb{R}$, how can one establish that $$F(a+b)=F(a)+f(a)b+ b^2\int_0^1 (1-s)f'(a+sb) \, ds,$$ which is part of a larger proof on page 508 of PDE Evans (2nd edition)?
Should I perform a change of variable to the second term on the RHS?

Taylor's formula with integral remainder: 
$$
F(a+b)=F(a) + F'(a) b + \int^{a+b}_a (a+b-t) F''(t)  \ dt
$$
Make a substitution $t=a+bs,$ and we get 
$$
F(a+b)=F(a) + F'(a) b + b^2\int^{1}_0 (1-s) F''(t) \ dt.
$$
Using $F(x)=\int^x_a f(w) \ dw,$ we get exactly the required result. 
