Confused with integral and natural logarithm When reading about ideal gas and adiabatic expansion, I got stuck with the following: W_(ab)= int_(V_a)^(V_b)(dV)/(V^alpha)=(V_b^(1- alpha)-V_a^(1- alpha))/(1-alpha)

Martin Hart 2022-10-13 Answered
Confused with integral and natural logarithm
When reading about ideal gas and adiabatic expansion, I got stuck with the following:
W a b = V a V b d V V α = V b 1 α V a 1 α 1 α
I know the following rule, but I couldn't come to the above:
V a V b d V V = l n | V b V a |
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Answers (1)

inmholtau5
Answered 2022-10-14 Author has 16 answers
You can write it as
V a V b V α   d V
It is obvious, that a 1 has to be added to the exponent to get the integral. If you differentiate a polynomial, the exponent decreases by 1.
V a V b V α   d V = b V 1 α | V a V b
b is a unknown factor.
If you differentiate b V 1 α you get b ( 1 α ) V α
Thus the factor b has to be 1 1 α
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