The intuitive derivation for the integrated form of the Kolmogorov backward equation makes sense and is given in the Act Ed notes on page 17 and 18. However, if I use the same logic when deriving the integrated form of the Kolmogorov forward equation I get a different equation from the notes.

My attempt is uploaded here: http://imgur.com/hGzy1fu

I am trying to follow the logic used in the backward case, noting that I change the focus from the first jump to the last jump.

Please help