Question

решите ВСЕГО один интеграл

Answer 1

$\int \frac{7-4x}{(5x+2)(x-2)}\, dx=\int \frac{A}{5x+2}\, dx+\int \frac{B}{x-2}\, dx=Q\\\\\frac{7-4x}{(5x+2)(x-2)}=\frac{A}{5x+2}+\frac{B}{x-2}\\\\7-4x=A(x-2)+B(5x+2)\\\\x=2:\; \; B=\frac{7-4x}{5x+2}=\frac{7-8}{10+2}=-\frac{1}{12}\; ;\\\\x=-\frac{2}{5}:\; \; A=\frac{7-4x}{x-2}=\frac{7-8/5}{-2/5-2}=-\frac{43}{12}\\\\Q=-\frac{43}{12}\int \frac{dx}{5x+2}-\frac{1}{12}\int \frac{dx}{x-2}=-\frac{43}{12\cdot 5}\cdot ln|5x+2|-\frac{1}{12}\cdot ln|x-2|+C$