If \( \int_{0}^{2 \pi} \frac{d x}{2+\sin 2 x}=k \int_{0}^{\frac{\pi...

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If \( \int_{0}^{2 \pi} \frac{d x}{2+\sin 2 x}=k \int_{0}^{\frac{\pi}{2}} \frac{d t}{3+\cos ^{2} t} \), then the value \( k \), is:
\( \mathrm{P} \)
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(1) 4
(2) 8
(3) 12
(4) 16
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