Total Pressure and Total Momentum

Bernoulli’s equation indicates that

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:
$P_o = frac{1}{2}rho_1 u_1^2 + P_1 = frac{1}{2}rho_2 u_2^2 + P_2$

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However, when you use conservation of momentum integral equation, you get

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$rho_1 u_1^2 A_1 + P_1 A_1 = rho_2 u_2^2 A_2 + P_2 A_2$

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For A1 = A2, this simplifies to:

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$rho_1 u_1^2 + P_1 = rho_2 u_2^2 + P_2$

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Why doesn’t this equation match up with Bernoulli’s equation i.e. where is the 1/2 term going?