If \(\displaystyle \int_0^{\pi/2} \dfrac{dx}{a^2\cos^2 x + b^2\sin^2 x}=\frac{\pi}{2ab}\), then find the value of \(\displaystyle \int_0^{\pi/2} \dfrac{dx}{\left(4\cos^2 x + 9\sin^2 x\right)^2}\).

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