Using the substitution \(u=\tan\dfrac{x}{2}\), show that \(\displaystyle\int\dfrac{1}{2+\cos x}\,\mathrm{d}x=\int\dfrac{2}{3+u^2}\,\mathrm{d}u\). Hence, find the exact value of \(\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{1}{2+\cos x}\,\mathrm{d}x\).[6]
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