Задача №1570

\[ \frac{x}{2x^2-3x-2} =\frac{\frac{x}{2}}{\left(x+\frac{1}{2}\right)(x-2)} =\frac{A}{x+\frac{1}{2}}+\frac{B}{x-2} =\frac{A(x-2)+B\left(x+\frac{1}{2}\right)}{\left(x+\frac{1}{2}\right)(x-2)} \]

\[ \frac{x}{2}=A(x-2)+B\left(x+\frac{1}{2}\right);\\ \begin{aligned} & x=-\frac{1}{2};\;-\frac{1}{4}=-\frac{5}{2}A;\;A=\frac{1}{10}.\\ & x=2;\;1=\frac{5}{2}B;\;B=\frac{2}{5}. \end{aligned} \]

\[ \int\frac{xdx}{2x^2-3x-2} =\int\left(\frac{1}{10}\cdot\frac{1}{x+\frac{1}{2}}+\frac{2}{5}\cdot\frac{1}{x-2}\right)dx =\frac{1}{10}\ln\left|x+\frac{1}{2}\right|+\frac{2}{5}\ln|x-2|+C \]