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Without explicit knowledge of the functions $v,w,p$ or some relationships between them you cannot evaluate this integral. With explicit knowledge (or relationships) you can apply the rules $$ dw(p(x))=w'(p(x))\,p'(x)\,dx\,,\quad\quad dw(1-p(x))=-w'(1-p(x))\,p'(x)\,dx\, $$ to see if you get something that can be integrated. I gave both formulas because your ...

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