$\int\limits_{0}^{1}\frac{2x+5}{x^2-2x-5}dx$$=\int\limits_{0}^{1}\frac{2x-2}{x^2-2x-5}dx+7\int\limits_{0}^{1}\frac{dx}{x^2-2x-5}$
$=\int\limits_{0}^{1}\frac{d(x^2-2x-5)}{x^2-2x-5}+7\int\limits_{0}^{1}\frac{dx}{(x-1)^2-6}$
$=ln|x^2-2x-5||_0^1+\frac{7}{2\sqrt6}ln|\frac{x-1-\sqrt6}{x-1+\sqrt6}||_0^1$
$=ln\frac{6}{5}-\frac{7}{2\sqrt6} ln\frac{1+\sqrt6}{\sqrt6-1}$
Ở trên ta đã dùng hai nguyên hàm cơ bản
$\int\frac{du}{u}=ln|u|$
$\int\frac{du}{u^2-c}=\frac{1}{2\sqrt c}ln|\frac{u-c}{u+c}|$