Let \(C\) be the region enclosed by the \(x\)-axis and the two circles \(x^2 + y^2 = 1\) and \(x^2+y^2 = 4\) (as shown by the red curves in the figure).

What is the value of \[ \displaystyle \oint_C (y^2\, dx + 5xy\, dy) ? \]

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