Задача №1569

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

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

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