Question

Интеграл от пи/3 до 0: sindxинтеграл от 1 до 0: dx/x+2интеграл от √3 до 1: dx/1+x^2​

Answer 1

$\int\limits^{\frac{\pi}{3}}_0 {sinx} \, dx =(-cosx) \bigg |_0^\frac{\pi}{3}=-cos\frac{\pi}{3}-(-cos0)=-cos\frac{\pi}{3}+cos0=-\frac{1}{2} +1=\frac{1}{2}\\ \\\\ \int\limits^1_0 {\frac{1}{x+2} } \, dx =(ln|x+2|) \bigg |^1_0=ln|1+2|-ln|0+2|=ln3-ln2=ln\frac{3}{2} \\ \\ \\ \int\limits^{\sqrt{3}}_1 {\frac{1}{1+x^2}} \, dx =arctgx \bigg |^{\sqrt{3}}_1=arctg\sqrt{3}-arctg1=\frac{\pi}{3}-\frac{\pi}{4}=\frac{4\pi-3\pi}{12}=\frac{\pi}{12}$