ccałka

[haharuka]

\(\int_{}^{} \frac{x}{2x^2+2x+3}\)

[eresh]

\(\int\frac{xdx}{2x^2+2x+3}=\frac{1}{4}\int\frac{4x+2-2}{2x^2+2x+3}=\frac{1}{4}\int\frac{4x+2}{2x^2+2x+3}dx-\frac{1}{2}\int\frac{dx}{2(x^2+x+\frac{3}{2})}=\\=\frac{1}{4}\ln|2x^2+2x+3|-\frac{1}{2}\int\frac{dx}{(x+\frac{1}{2})^2+\frac{5}{4}}=\frac{1}{4}\ln|2x^2+2x+3|-\frac{1}{2}\int\frac{dx}{(x+\frac{1}{2})^2+(\frac{\sqrt{5}}{2})^2}=\\
=\frac{1}{4}\ln|2x^2+2x+3|-\frac{1}{2}\cdot \frac{2}{\sqrt{5}}\arctg\frac{x+\frac{1}{2}}{\frac{\sqrt{5}}{2}}=\\
=\frac{1}{4}\ln|2x^2+2x+3|- \frac{1}{\sqrt{5}}\arctg\frac{2x+1}{\sqrt{5}}\)