How do I transform Integral H(x)dF(x) into something like Integral H'(x)F(x)dx ? What is the rule and why?

Answer 1

It's called the integration by parts formula.
Int u dv=uv- int v du

Both u and v need to be absolutely continuous for this to work. A continous derivative for both is plenty.

Answer 2

d( H(x) F(x) ) = H(x) dF(x) + dH(x) F(x)

or

( H(x) F(x) )' = H(x) F'(x) + H'(x) F(x)

So if you can integrate any two of (H(x) F(x))' dx, H(x) F'(x) dx, or H'(x) F(x) dx, then you can find the third by adding/subtracting the two known ones (but do get the order correct for the subtraction!). And the integral of (H(x) F(x))' dx is just H(x) F(x), so you are already spotted one! :-)

Answer 3

(u dv) = uv - ( v du )

or F(x) H'(x) dx = F(x) H(x) - H(x) F'(x) dx