• Metodo dei rettangoli

${\int }_0^1\sqrt{\frac{1-x}{1+x}}dx=\frac{1}{10}\left[f\left(0\right)+f\left(0.1\right)+f\left(0.2\right)+f\left(0.3\right)+f\left(0.4\right)+f\left(0.5\right)+f\left(0.6\right)+f\left(0.7\right)+f\left(0.8\right)+f\left(0.9\right)\right]=$

$=\frac{1}{10}\left[\sqrt{\frac{1-0}{1+0}}+\sqrt{\frac{1-0.1}{1+0.1}}+\sqrt{\frac{1-0.2}{1+0.2}}+\sqrt{\frac{1-0.3}{1+0.3}}+\sqrt{\frac{1-0.4}{1+0.4}}+\sqrt{\frac{1-0.5}{1+0.5}}+\sqrt{\frac{1-0.6}{1+0.6}}+\sqrt{\frac{1-0.7}{1+0.7}}+\sqrt{\frac{1-0.8}{1+0.8}}+\sqrt{\frac{1-0.9}{1+0.9}}\right]\simeq 0.6169667033$
$\pi =2\left(1+{\int }_0^1\sqrt{\frac{1-x}{1+x}}dx\right)\simeq 3.2339334065$

• Metodo dei trapezi

${\int }_0^1\sqrt{\frac{1-x}{1+x}}dx=\frac{1}{10}\left[\frac{f\left(0\right)+f\left(1\right)}{2}+f\left(0.1\right)+f\left(0.2\right)+f\left(0.3\right)+f\left(0.4\right)+f\left(0.5\right)+f\left(0.6\right)+f\left(0.7\right)+f\left(0.8\right)+f\left(0.9\right)\right]=$
$=\frac{1}{10}\left[\frac{\sqrt{\frac{1-0}{1+0}}+\sqrt{\frac{1-1}{1+1}}}{2}+\sqrt{\frac{1-0.1}{1+0.1}}+\sqrt{\frac{1-0.2}{1+0.2}}+\sqrt{\frac{1-0.3}{1+0.3}}+\sqrt{\frac{1-0.4}{1+0.4}}+\sqrt{\frac{1-0.5}{1+0.5}}+\sqrt{\frac{1-0.6}{1+0.6}}+\sqrt{\frac{1-0.7}{1+0.7}}+\sqrt{\frac{1-0.8}{1+0.8}}+\sqrt{\frac{1-0.9}{1+0.9}}\right]\simeq 0.5669667033$
$\pi =2\left(1+{\int }_0^1\sqrt{\frac{1-x}{1+x}}dx\right)\simeq 3.1339334065$

• Metodo delle parabole

${\int }_0^1\sqrt{\frac{1-x}{1+x}}dx=\frac{1}{30}\left\{f\left(0\right)+f\left(1\right)+2\left[f\left(0.2\right)+f\left(0.4\right)+f\left(0.6\right)+f\left(0.8\right)\right]+4\left[f\left(0.1\right)+f\left(0.3\right)+f\left(0.5\right)+f\left(0.7\right)+f\left(0.9\right)\right]\right\}=$
$=\frac{1}{10}\left\{\sqrt{\frac{1-0}{1+0}}+\sqrt{\frac{1-1}{1+1}}+2\cdot \left[\sqrt{\frac{1-0.2}{1+0.2}}+\sqrt{\frac{1-0.4}{1+0.4}}+\sqrt{\frac{1-0.6}{1+0.6}}+\sqrt{\frac{1-0.8}{1+0.8}}\right]+4\cdot \left[\sqrt{\frac{1-0.1}{1+0.1}}+\sqrt{\frac{1-0.3}{1+0.3}}+\sqrt{\frac{1-0.5}{1+0.5}}+\sqrt{\frac{1-0.7}{1+0.7}}+\sqrt{\frac{1-0.9}{1+0.9}}\right]\right\}\simeq 0.568990032$
$\pi =2\left(1+{\int }_0^1\sqrt{\frac{1-x}{1+x}}dx\right)\simeq 3.1379800641$