Question

интеграл 1/(2x^2+x+2)dx

Answer 1

Выделяем полный квадрат
$\int \frac{dx}{2 \cdot (x^{2} +\frac{x}{2}+1)}=\frac{1}{2} \cdot \int \frac{dx}{(x^{2} +2 \cdot \frac{1}{4} \cdot x+\frac{1}{16})- \frac{1}{16} +1}=\frac{1}{2} \cdot \int \frac{dx}{(x+\frac{1}{4})^{2}+ \frac{15}{16}}=$$\frac{1}{2} \cdot \int \frac{dx}{(x+\frac{1}{4})^{2}+ (\sqrt{\frac{15}{16}})^{2}}=\frac{1}{2} \cdot \frac{1}{\sqrt{\frac{15}{16}}} \cdot arctg( \frac{x+\frac{1}{4}}{\sqrt{\frac{15}{16}}} )+C= \frac{2}{\sqrt{15}} arctg( \frac{4x+1}{ \sqrt{15} } )+C$