Решите \(\int{\frac{2x+5}{x^2-9}}\)

Решение

\(\int{\frac{2x+5}{x^2-9}}=\int{\frac{11}{6(x-3)}+\frac{1}{6(x+3)}dx}=\frac{11}{6}\int{\frac{1}{x-3}dx}+\frac{1}{6}\int{\frac{1}{x+3}dx}=[t_1=x-3,t_2=x+3]=\frac{1}{6}\int{\frac{1}{t_1}dt_1}+\frac{1}{6}\int{\frac{1}{t_2}dt_2}=\frac{11}{6} ln|t_1|+\frac{1}{6}ln|t_2|=\frac{11}{6}ln|x-3|+\frac{1}{6}ln|x+3|\) +C

Ответ: \(\frac{11}{6}ln|x-3|+\frac{1}{6}ln|x+3|\) +C.