[tex]f(x) = \dfrac{2x+3}{x^2+3x+3} \\ \\ \int\limits^2_1 {(x^2+3x+3)\cdot f(x)} \, dx = \int\limits^2_1 {(x^2+3x+3)\cdot \dfrac{2x+3}{x^2+3x+3} } \, dx= \\ \\ = \int\limits^2_1 {(2x+3)} \, dx = \Big(2\cdot\dfrac{x^2}{2}+ 3x\Big)\Big|_1^2= \Big(x^2+ 3x\Big)\Big|_1^2 = \\ \\ =2^2+3\cdot 2-(1^2+3\cdot 1) = 4+6-1-3 = 6[/tex]
