Example 3. Evaluate the integral $I=\int_{-\pi / 2}^{\pi / 2} \frac{2 d x}{7-3 \cos 2 x}$.
Solution. If $x \in\left(-\frac{\pi}{2} ; \frac{\pi}{2}\right)$, then $-1<\cos 2 x<1$, and therefore $4<7-3 \cos 2 x<10$. Consequently, $\frac{1}{5}<\frac{2}{7-3 \cos 2 x}<\frac{1}{2}$. Applying property 6.3) of the definite integral with $m=\frac{1}{5}, M=\frac{1}{2}, b-a=\pi$, we obtain the estimate $\frac{\pi}{5}<I<\frac{\pi}{2}$.

Answer. $\frac{\pi}{5}<I<\frac{\pi}{2}$.