using the isentropic equation
pV^k = constant = p1V1^k
you can write V in function of p so you get another integral
int (p1/p)^(1/k)*V1 dp
=
p1^(1/k)*V1 * int p^(-1/k) dp
An analog conversion is possible for the pdV, but i found both as methode to calculate the required energy. Now I...