Example 2.18. Estimate the upper and lower bounds of the value of the integral $\int_{0}^{\pi / 6} \frac{d x}{1+3 \sin ^{2} x}$.
Solution. $\frac{1}{1+3 \sin ^{2}(\pi / 6)} \leqslant \frac{1}{1+3 \sin ^{2} x} \leqslant \frac{1}{1+3 \sin ^{2} 0}$.

From this, $\frac{2 \pi}{21} \leqslant \int_{0}^{\pi / 6} \frac{d x}{1+3 \sin ^{2} x} \leqslant \frac{\pi}{6}$.