\[\large A = \displaystyle \int_0^{{\pi} / {2}} \frac{\psi (x)}{\psi (x) + \psi \left(\frac{\pi}{2} - x \right)} \, dx \]

Compute \(\lfloor A \times 1000 \rfloor\).

For your final step of your calculation, use the approximation \( \pi = \dfrac{22}7 \).

**Notations**:

\( \lfloor \cdot \rfloor \) denotes the floor function.

\( \psi (\cdot) \) denotes the Digamma function.

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