Clarification of Wiki Proof of Taylor's Theorem in One Variable

Question

I am having difficulty with a particular line of following proof of Taylor's Theorem.

The line I am having difficulty with is the "Another application yields". So I gather that the author is using integration by parts on $\int_{a}^{x}(x-t)f^{''}(t)dt$.

So let $u=f^{''}(t)$ and $dv=(x-t)$, then I get that

$\int_{a}^{x}(x-t)f^{''}(t)dt = (xt-t^2/2)f^{''}(t)|^{x}_{a} - \int_a^{x}(xt-t^2/2)f^{'''}(t)dt $

$= \frac {(x-a)^2}{2} - \int_a^{x}(xt-t^2/2)f^{'''}(t)dt$,

but I am not sure how to proceed to obtain the next line. Any clarifications/simpler ways of seeing this much appreciated.

Answer 1

$$\int_a^x(x-t)f^"(t)dt = \int_a^xf^"(t)d(-\frac12(x-t)^2)$$ $$=(-\frac12(x-t)^2)f^"(t)|^{x}_{a} -\int_a^xf^{'''}(t)(-\frac12(x-t)^2)dt$$ $$=\frac12(x-a)^2)f^"(a)+\int_a^xf^{'''}(t)(\frac12(x-t)^2)dt$$