Задача №1510
Условие
Найти интеграл \(\int\frac{xdx}{\sqrt{3x^2-11x+2}}\).
Решение
\[
\int\frac{xdx}{\sqrt{3x^2-11x+2}}
=\int\frac{\frac{1}{6}\cdot(6x-11)+\frac{11}{6}}{\sqrt{3x^2-11x+2}}=\\
=\frac{1}{6}\cdot\int\left(3x^2-11x+2\right)d\left(3x^2-11x+2\right)+\frac{11}{6\sqrt{3}}\cdot\int\frac{d\left(x-\frac{11}{6}\right)}{\sqrt{\left(x-\frac{11}{6}\right)^2-\frac{97}{36}}}=\\
=\frac{1}{3}\cdot\sqrt{3x^2-11x+2}+\frac{11}{6\sqrt{3}}\ln\left|x-\frac{11}{6}+\sqrt{x^2-\frac{11}{3}x+\frac{2}{3}}\right|+C.
\]
Ответ:
\(\frac{1}{3}\cdot\sqrt{3x^2-11x+2}+\frac{11}{6\sqrt{3}}\ln\left|x-\frac{11}{6}+\sqrt{x^2-\frac{11}{3}x+\frac{2}{3}}\right|+C\)